A bag contains yellow, blue and red counters. Asif takes one at random. \(P(\text{yellow})=27\%\), \(P(\text{blue})=\frac35\).

Find the probability of red. [2]

Marwa has 10 numbered tiles: \(1,2,2,3,3,4,4,4,4,4\).

(a) She chooses one tile at random, replaces it, then chooses another. Find the probability that both have the same even number. [3]

(b) She instead chooses three tiles at random without replacement. Find the probability that their numbers sum to 11. [3]

Juan has 2 red, 3 green and 5 blue counters in a bag. He chooses two at random without replacement. Find the probability that their colours differ. [4]

Reem spins a spinner numbered 1 to 6 on 50 occasions, obtaining these results.

(a) State the relative frequency of getting 4. [1]

(b) Estimate how many 4s she will get in another 200 spins. [1]

The probability of selecting a red counter from a bag is 0.24.

(a(i)) Find the probability of not selecting red. [1]

(a(ii)) Sunil selects a counter, replaces it, then selects another. Find the probability that exactly one is red. [2]

(b) A box contains 50 pens, of which n are blue. Ahmed selects two without replacement. The probability both are blue is \(3/35\). Find n. [3]

Pryanka can win, lose or draw a game.

(a) Complete the missing probability. [2]

(b) She plays 120 games. Find the expected number she wins. [1]

The probability of picking a green pen is 0.17. Find the probability of not picking a green pen. [1]

(a) A bag contains 6 red, 3 green and 1 blue marble. Two marbles are picked at random with replacement. Find the probability both are green. [2]

Another bag contains 4 red and 2 yellow counters. Two are picked at random without replacement.

(b(i)) Complete the tree diagram. [2]

(b(ii)) Find the probability that one of the two counters is yellow. [3]

Seven cards are numbered 1, 2, 3, 4, 5, 6 and 7. Samira picks a card at random and replaces it.

(a) Find the probability of an odd number. [1]

(b) She repeats this 35 times. Find the expected number of odd numbers. [1]

Angela picks one number at random from 1, 2, 3, then one from 4, 5, 6, and adds them.

(a) Complete the possible-outcomes table. [2]

(b) Given the total is odd, find the probability that one of her numbers is 3. [2]

The probability of sunny weather on any day is 0.7.

(a) Find the probability that it is not sunny. [1]

(b) Rohit walks with probability 0.9 when it is sunny and 0.2 when it is not sunny. Find the probability he walks on any day. [3]

The Venn diagram represents 33 students: G study geography and H study history.

(a(i)) A student is chosen at random. Find the probability they study neither subject. [1]

(a(ii)) Find the probability they study both subjects. [1]

(b) Two history students are chosen at random. Find the probability one also studies geography and the other does not. [3]

Bags A and B contain red and yellow sweets only. Stephan picks one sweet at random from each bag. The probability of red from A is \(3/5\), and the probability of red from both bags is \(2/15\).

(a) Find the probability of red from B. [2]

(b) Find the probability of yellow from both bags. [3]

Sarah rolls a fair six-sided die twice. Find the probability that both results exceed 4. [2]

Bag A has 5 white and 3 black balls; bag B has 3 white and 1 black ball.

(a) Two balls are picked from B without replacement. Find the probability both are black. [1]

The balls are returned to B. Kyle then picks one ball from each bag.

(b(i)) Complete the probability tree. [2]

(b(ii)) Find the probability both balls are the same colour. [3]

(c) Return the balls to their bags. Jo picks one ball from A and transfers it to B, then picks from B. Find the probability her pick from B is black. [3]

A club has 100 members: 60 swim (S), 70 cycle (C), and 25 do neither.

(a) Complete the Venn diagram. [3]

(b(i)) Find \(P(C)\) for one member chosen at random. [1]

(b(ii)) Find \(P(S\cap C)\). [1]

(b(iii)) Find \(P(S\cup C\prime)\). [1]

Mahir picks uniformly from 5,10,15, then uniformly from 4,5,6, and adds the two numbers.

(a) Complete the sample space table. [1]

(b) Given the sum is odd, find the probability that one chosen number is 15. [2]

Flowers in a vase are red, pink or white. Ruth chooses a flower at random. P(not red)=0.9 and P(not pink)=0.65. Find P(white). [2]

A three-sided spinner has labels A, B and C, with P(C)=0.35. Farid spins it 40 times. Find the expected number landing on C. [1]

A bag contains 2 green, 5 red and 6 blue buttons. Two are taken at random without replacement. Find the probability of different colours. [4]

In a survey of 40 employees, 3 walk to work. The company has 1240 employees. Find the expected number who walk to work. [2]

The probability of rain today is 0.15. Find the probability it will not rain. [1]

Geetha chooses a toy at random. P(wooden)=0.6.

(a) Find P(not wooden). [1]

(b) The box contains only wooden, plastic and metal toys. Complete the table. [2]

Bags A and B contain only red and blue counters. Stephan chooses from A and Jen from B. P(Stephan red)=0.4; P(both red)=0.25. Find P(both blue). [4]

Seven cards show N,A,M,I,B,I,A.

(a(i)) A card is chosen at random. Find P(H). [1]

(a(ii)) Find P(B). [1]

(b) A card is chosen, replaced, then a second is chosen. Find P(both I). [2]

(c(i)) Two cards are chosen without replacement. Find P(one I and one N). [3]

(c(ii)) Find P(different letters). [3]

(d) Nina draws without replacement until the first A. The probability it occurs on card n is 4/21. Find n. [2]

Jacinda can win, lose or draw a game. The probability she wins is 0.28.

(a) She is twice as likely to draw as to lose. Find the probability she loses. [2]

(b) She plays 150 times. Find the expected number of wins. [1]

75 people are asked whether they have a car C and a job J.

A person is chosen from those who have a car. Find the probability they also have a job. [1]

A box contains 3 blue and 5 red pens.

(a) Mia picks one pen at random. Find the probability it is red. [1]

She replaces it, then picks two pens with replacement.

(b(i)) Complete the tree. [2]

(b(ii)) Find the probability the two pens have the same colour. [3]

(c) She now picks three pens without replacement. Find the probability of two blue and one red. [3]

A bag has 24 beads including 10 yellow, with the others red or blue. One is chosen at random.

(a(i)) Find P(yellow). [1]

(a(ii)) Find P(not yellow). [1]

Another bag has 5 green,6 white and 4 black marbles. Meera chooses two without replacement.

(b(i)) Find P(first black and second white). [2]

(b(ii)) Find P(different colours). [4]

A bag contains 5 white and 3 black balls.

(a(i)) One marble is selected at random. Find P(white). [1]

(a(ii)) It is replaced. This is repeated 120 times. Find the expected number of white balls selected. [1]

Two balls are now selected at random with replacement.

(b(i)) Find P(same colour). [3]

(b(ii)) Find P(different colours). [1]

(c) Three balls are selected without replacement. Find P(exactly two white and one black). [3]

Jamie hits a target with probability \(1/3\) on each independent attempt. The probability of his first hit occurring on his \(n\)th attempt is \(64/2187\).

Find \(n\). [2]

A bag contains 5 green, 2 blue and 6 white buttons. Maya takes two at random without replacement.

Find the probability that one is green and the other is not green. [3]

Eric uses four paint colours with the following probabilities.

Find \(x\). [2]

Bags \(A,B\) each contain red and yellow sweets. Anna picks from \(A\) and Ben independently picks from \(B\). \(P(\text{Anna red})=2/5\) and \(P(\text{both yellow})=1/10\).

Find the probability that both pick a red sweet. [3]

A spinner has four possible outcomes.

Complete the table. [2]

A five-sided spinner has the following probabilities.

(a) Complete the table. [2]

(b) Dan spins once. Find the probability of red or blue. [2]

Rama records how students travel to school.

(a) Complete the missing probability. [2]

(b) There are 1800 students. Find the expected number who walk. [1]

Lucia has two fair spinners. A has five sides labelled 1, 2, 3, 4, 5. B has nine sides labelled 3, 3, 3, 4, 4, 4, 4, 5, 5. She records whether each spin gives a prime number.

(a(i)) Complete the tree diagram; the first prime branch for A is \(3/5\). [2]

(a(ii)(a)) Find the probability both numbers are prime. [2]

(a(ii)(b)) Find the probability the numbers are not both prime. [1]

(b) A is spun 120 times. Find the expected number of prime results. [1]

(c) B is spun twice. Find the probability the sum is 9 or more. [3]

(d) B is spun until a 4 occurs. Find the probability the first 4 is on the nth spin. [2]

Two fair six-sided dice have labels: A: \(1,2,2,2,3,6\); B: \(2,3,3,4,4,4\).

(a(i)) A is rolled once. Find the probability of 6. [1]

(a(ii)) A is rolled 150 times. Find the expected number of 6s. [1]

(b(i)) Roll A and B once each. Find the probability the total is 6. [3]

(b(ii)) Given the total is 6, find the probability both numbers are 3. [2]

(c) B is rolled until it first gives a 3 on the nth roll. This probability is \(32/729\). Find \(n\). [2]

(a(i)) Shalini is late for school with probability \(1/6\) on any day. Complete the Monday–Tuesday tree. [2]

(a(ii)) Find the probability she is late Monday and not late Tuesday. [2]

(b(i)) Use set notation for the region containing only one student. [1]

(b(ii)) Find \(n(T^\prime\cap(G\cup M))\). [1]

(b(iii)) Pick one of the 50 students. Find the probability of trainers but no glasses. [1]

(b(iv)) Pick two students at random from those wearing trainers. Find the probability both have phones. [3]

The probability of a red sweet is 0.05. Find the probability of not red. [1]

A random card has one of these shapes. Complete the probability table. [2]

Eleven letter cards: \(\mathrm{V,E,N,N,D,I,A,G,R,A,M}\).

(a(i)) Describe the shaded region in set notation. [1]

(a(ii)) Shade \(Q\cup P^\prime\) on the first diagram and \((K\cup L)\cap M^\prime\) on the second. [2]

(b(i)) Choose one letter card at random. Find the probability it is not A. [1]

(b(ii)) Choose a card, replace it, then choose another. Find the probability exactly one is N. [3]

(c(i)) Of 50 students, 3 like neither English nor mathematics, 33 like English and 42 like mathematics. Complete a Venn diagram with sets E and M. [2]

(c(ii)) Select one student. Find the probability they like both. [1]

(c(iii)) Select two students without replacement. Find the probability both like mathematics. [2]

(c(iv)) Select two from those liking English. Find the probability both also like mathematics. [2]

(a) Of 500 people questioned at a festival, 380 are camping. The festival has 55300 people. Estimate the total number camping. [2]

(b) Of 12 friends, 5 travel by car, 4 by bus and 3 by train. Choose two friends at random. Find the probability they travel by different types of transport. [4]

(c) Arno pays \$43.68 for a student ticket, saving 16% on full price. Find the full price. [2]

(d(i)) A football match attendance is 29800, to the nearest 100. People leave at 400 per minute, to the nearest 50. Find the lower bound for the time needed for everyone to leave. [3]

(d(ii)) A cricket match attendance is 27500, to the nearest 100. Find the upper bound for the difference in attendances. [2]

A bag contains 5 red, 4 blue and 3 green balls.

(a(i)) Megan chooses one ball. Find the probability it is red or blue. [1]

(a(ii)) She replaces it, then makes 60 independent draws with replacement. Find the expected red-or-blue count. [1]

(b) Mick chooses two balls without replacement. Find the probability their colours differ. [4]

(c) Marie draws without replacement until the first green ball. This happens on draw n with probability \(21/220\). Find n. [2]

Katy picks uniformly from 2, 3, 5 and then from 5, 6, 7, 9. Given that the sum is even, find the probability exactly one picked number is 5. [3]

Spinner probabilities: green \(2/5\), black \(1/4\), red unknown.

(a) Find the red probability. [2]

(b) Chang spins 120 times. Find the expected green count. [1]

Six cards are labelled −3, −2, 2, 3, 5 and 7. Regan chooses cards at random without replacement.

(a(i)) Two cards are multiplied. Find the probability the score is 35. [3]

(a(ii)) Find the probability the score is positive. [3]

(b) Three cards are added. Find the probability their total is 5. [4]

(a) 28 of 1000 voucher recipients used their voucher. How many of a further 16 500 recipients are expected to use it? [1]

(b(i)) A class of 15 has 7 red, 6 green and 2 white hats. Find the probability a randomly chosen student wears red. [1]

(b(ii)) Two students are selected without replacement. Show the probability of matching hat colours is \(37/105\). [3]

(b(iii)) Three students are selected without replacement. Find the probability at least two wear red. [4]

The probability of a train being late is 0.15. Find the probability that it is not late. [1]

A bag contains 3 blue, 8 white and 5 red counters. Two counters are chosen at random without replacement. Find the probability that both are red or both are white. [3]

Jane wins a game with probability \(7/10\).

(a) Find the probability that she does not win. [1]

(b) She plays 50 times. Find the expected number of wins. [1]

Malik buys bread daily. Probability of visiting in the morning is 0.7. Bread is available with probability 0.95 in the morning and 0.6 later. Find the probability that bread is available when he visits. [3]

The number of games lost is twice the number drawn.

Complete the relative-frequency table. [3]

Sachin randomly picks one of the first three multiples of 3, then one of the first three prime numbers. The score is their sum. All choices are equally likely.

(a) Complete the table of scores. [2]

(b) Given that the score is even, find the probability that one number picked is 9. [2]

Katy has 5 white flowers, x red flowers and \(2x+1\) yellow flowers. She picks one at random. The probability of white is \(1/12\). Find the probability of yellow. [4]

Sarah uses a fair four-sided spinner numbered 0, 1, 1, 3.

(a(i)) Which number is most likely? [1]

(a(ii)) She spins twice. Find the probability of 1 both times. [2]

(a(iii)) She spins until the first 3. The probability of first obtaining 3 on spin n is \(729/16384\). Find n. [2]

(b(i)) Scott must pass both a theory test and a practical test to pass an exam. The probabilities of passing are 0.9 and 0.8 respectively. Find the probability of passing the exam. [2]

(b(ii)) Find the probability of passing exactly one test. [3]

Suleika has six cards numbered 1 to 6.

(a(i)) She chooses one card at random and replaces it. Find the probability of 5 or 6. [1]

(a(ii)) She repeats this 300 times. Find the expected number of 5s or 6s. [1]

(b(i)) She chooses two cards at random without replacement. Find the probability that their sum is 5. [3]

(b(ii)) Find the probability that at least one card is a square number. [3]

Sofia has 8 blue and 7 red beads. She selects and replaces one bead 90 times. Find the expected number of red beads. [2]

A bag contains only blue, red, yellow and green balls. A ball is selected at random.

(a) Complete the probability table. [2]

(b) Abdul selects and replaces a ball 200 times. Find the expected number of red balls. [1]

A seed produces flowers with probability0.8. Given that it flowers, the colour is red with probability0.6, yellow with probability0.3, or another colour.

(a) Given that a seed flowers, find the probability that its flowers are neither red nor yellow. [1]

(b(i)) Complete the probability tree.

(b(ii)) Find the probability a seed produces red flowers. [2]

(b(iii)) Find the probability a seed produces neither red nor yellow flowers, including seeds that do not flower. [3]

(c) Two seeds are chosen independently. Find the probability exactly one produces flowers. [3]

On any Saturday, Arun plays football with probability3/4 and Bob with probability2/5. Treat their decisions as independent, and Saturdays as independent.

(a(i)) Complete the probability tree.

(a(ii)) Find the probability both play on one Saturday. [2]

(a(iii)) Find the probability exactly one plays on one Saturday. [3]

(b) Find the probability Bob plays on exactly2 of the next3 Saturdays. [3]

(c) Given Arun plays, he scores the winning goal with probability1/7. Find the probability he scores the winning goal on one Saturday. [2]

A bag contains7 red,5 green and2 pink discs.

(a) Helen selects and replaces a disc140 times. Find her expected number of green selections. [2]

(b) She adds9 green and some pink discs. Green probability is now2/7. How many pink discs were added? [2]

Of 32 students, 5 study no languages, 15 study German G and 18 study Spanish S.

(a(i)) Complete the Venn diagram. [2]

(a(ii)) Find the probability a random student studies Spanish but not German. [1]

(a(iii)) A German student is chosen. Find the probability they also study Spanish. [1]

(b) A bag has 54 red marbles and some blue marbles; 36% are red. Find the blue count. [2]

Another bag has 15 red and 10 yellow beads. Ariana selects, records and replaces a bead, then selects another.

(c(i)) Find the probability of two reds. [2]

(c(ii)) Find the probability of not selecting two reds. [1]

(d) A box has 15 red,8 yellow and2 green pencils. Two are selected without replacement. Find the probability at least one is red. [3]

Six discs have the following colours and numbers:

(a(i)) One disc is selected at random. Find the probability its number is4. [1]

(a(ii)) Find the probability it is red and numbered3. [1]

(a(iii)) Find the probability it is blue and numbered4. [1]

(b(i)) Two discs are selected without replacement. Find the probability both are numbered3. [2]

(b(ii)) Find the probability both have the same colour. [3]

(c) Two discs are selected with replacement. Find the probability both have the same colour. [3]

The eleven letter tiles are P,O,S,S,I,B,I,L,I,T,Y. Morgan selects without replacement.

(a(i)) Find the probability the first tile is Y. [1]

(a(ii)) Find the probability of B followed by Y. [2]

(a(iii)) Find the probability the first two letters are the same. [3]

Morgan now selects a third tile.

(b(i)) Find the probability all three letters are the same. [2]

(b(ii)) Find the probability exactly two of the three are the same. [5]

(b(iii)) Find the probability all three are different. [2]

A factory sweet has probability0.0028 of being the wrong shape. Of25000 sweets, how many are expected to be wrong? [2]

Sushila, Ravi and Talika each have a bag containing10 red and8 blue balls.

(a) Sushila selects one ball. Find the probability it is red. [1]

(b) Ravi selects two without replacement. Find the probability one is red and one blue. [3]

(c) Talika selects three without replacement. Find the probability all have the same colour. [4]

Five cards are numbered 1, 2, 3, 4, 5.

Two cards are taken at random without replacement. Find the probability that both show an even number. [2]

Andrei cycles to school with probability r.

(a) Express the probability he does not cycle in terms of r. [1]

(b) Benoit does not cycle with probability1.3−r; both do not cycle with probability.4.

(b(i)) Complete the equation in r: (…)×(…)=.4. [1]

(b(ii)) Show that it simplifies to \(10r^2-23r+9=0\). [3]

(b(iii)) Solve this equation by factorisation. [3]

(b(iv)) Find the probability that Benoit does not cycle. [1]

(a) Angelo’s bag contains3 white counters and x black counters. He draws two without replacement.

(a(i)) Complete the probability of two black counters: \(\frac{x}{x+3}\times(\ldots)\). [2]

(a(ii)(a)) This probability is7/15. Show that \(4x^2-25x-21=0\). [4]

(a(ii)(b)) Solve by factorisation. [3]

(a(ii)(c)) Give the number of black counters. [1]

(b) Esme draws3 counters without replacement from5 green and4 red. Find the probability all3 have the same colour. [4]

The school bus is late with probability 9/10. Seb uses it with probability 15/16 when late, and 3/4 when on time. Find the probability that Seb uses the bus. [3]

Harris passes his driving test at the first attempt with probability 0.6. If he fails, his probability of passing at each further attempt is 0.75.

(a) Find the probability that he passes at the second attempt. [2]

(b) Find the probability that he takes no more than three attempts. [2]

Donald chooses randomly from the five cards shown.

(a(i)) Find the probability of an even number of dots. [1]

(a(ii)) Find the probability of a prime number of dots. [1]

He chooses two cards without replacement.

(b(i)) Find the probability that the total dots is 5. [3]

(b(ii)) Find the probability that the total is odd. [3]

A bag has 4 red and 2 yellow marbles. Behnaz draws two without replacement.

(a(i)) Find the probability both are red. [2]

(a(ii)) Find the probability they are not both red. [1]

(b) A second bag has 5 blue and 2 green marbles. Bryn draws without replacement until getting a green marble. Find the probability of getting green on the first, second or third attempt. [4]

Samira and Sonia each have 20 sweets: 5 red, 6 green, 9 yellow.

(a) Samira chooses one randomly. Find the probability of yellow. [1]

Sonia chooses two without replacement.

(b(i)) Show the probability both are green is 3/38. [2]

(b(ii)) Find the probability the two are not both the same colour. [4]

A box has 3 blue, 4 red and 8 green pens only. One pen is chosen randomly.

(a) Find the probability that it is green. [1]

Another box has 7 black and 8 orange pens only. Two are chosen randomly without replacement.

(b) Find the probability of at least one orange pen. [3]

Kim wins a game with probability 0.72. Estimate how many of 225 games she will win. [2]

Boxes A and B contain blue and green pens only. Raphael randomly selects from A and Paulo from B. \(P(\text{Raphael blue})=\frac23,\quad P(\text{both blue})=\frac8{15}\).

(a) Find the probability Paulo chooses blue. [2]

(b) Find the probability both choose green. [3]

\(P(\text{rain})=\frac58,\quad P(\text{walk}\mid\text{rain})=\frac16,\quad P(\text{walk}\mid\text{no rain})=\frac7{10}\).

(a) Complete the tree. [3]

(b) Find the probability of rain and Rafael walking to school. [2]

(c) Find the probability that Rafael does not walk. [3]

(a(i)) Jojo chooses two cards randomly from A without replacement. Find the probability of matching numbers. [3]

(a(ii)) Jojo replaces them. Kylie chooses one card from A and one from B randomly. Find the probability of matching numbers. [3]

(a(iii)) Who is more likely to choose matching numbers? Show working. [1]

Set C has cards4,4,5,5,5. Lena chooses three randomly without replacement.

(b) Find the probability that the third card is4. [3]

200 people were asked which city they wished to visit next.

(a) Choose one randomly. Find the probability of Paris or Tokyo. [2]

(b) Choose two randomly. Find the probability one wishes to visit London and the other New York, as a percentage. [3]

20 chip packets:6 barbecue,10 salt,4 chicken. All selections are random without replacement.

(a(i)) For two packets, show P(both salt)=9/38. [2]

(a(ii)) Find P(two different flavours). [4]

(b) For three packets, find P(at least two chicken). [3]

Bag A:3 black,2 white; Bag B:1 black,3 white.

(a(i)) Choose one from each bag. Show black is more likely from A. [1]

(a(ii)) Find P(different colours). [3]

Return balls to their original bags; take 3 from A without replacement.

(b(i)) Find P(all black). [2]

(b(ii)) Find P(all white). [1]

(c) Return balls again. Take one from A, record its colour, transfer it to B, then take one from B. Find P(the second ball differs in colour from the first). [3]

Ravi spins a biased five-sided spinner.

(a) Find x. [3]

(b) For one spin, find P(2 or 3). [2]

(c(i)) For two independent spins, find P(2 both times). [2]

(c(ii)) Find P(sum of the two numbers is 3). [3]

(d) In 72 spins, how many 1 s are expected? [1]

The probability Stephanie wins her next tennis match is 0.85. Find the probability she does not win. [1]

The probability Pedro scores a goal in any match is \(\frac25\). Calculate the probability he scores in each of his next two matches. [2]

A fair spinner has five equal sectors labelled 1,3,3,4,6. Anna spins twice and adds the scores.

(a) Complete the table of totals. [1]

(b) Write the most likely total. [1]

(c(i)) Find the probability of a total less than 6. [2]

(c(ii)) Find the probability of a total of 3. [1]

Simon picks a card at random from each of two boxes. In one box each card has a triangle or a square; in the other each card is red or blue. The probability of a triangle is t and of red is r.

Complete the table in terms of r and t. [3]

Each morning P(rain)=2/3. P(Asha walks | rain)=1/7; P(Asha walks | no rain)=4/7.

(a) Complete the tree. [2]

(b) Find P(rain and walks). [2]

(c(i)) Find P(does not walk). [3]

(c(ii)) Find the expected number of days she does not walk in a term of 70 days. [2]

(d) Find the probability it rains on exactly one morning in 5 school days. [2]

An 80-bead bag has red and green beads; P(green)=0.35.

(a(i)) Find the number of red beads. [2]

(a(ii)) Marcos draws with replacement 240 times. Find the expected green count. [1]

Another bag has 2 blue,3 yellow,4 white marbles. Huma draws three times with replacement.

(b(i)) Find P(all yellow). [2]

(b(ii)) Find P(all different colours). [3]

(c) A bag has 2 green and 3 pink counters. Teresa draws three without replacement. Find P(more pink than green). [4]

Fair dice A,B have faces A:0,0,1,1,1,3 and B:1,1,2,2,2,3.

(a) Find P(A does not score 3). [1]

(b) Find P(A scores 0 twice). [2]

(c) Estimate the zero count in 60 rolls of A. [1]

(d(i)) Complete the product possibility diagram. [2]

(d(ii)(a)) Find P(product=2). [1]

(d(ii)(b)) Find P(product>3). [1]

(e) Eva rolls B until she scores 1. Find P(this first occurs on roll 5). [2]

Dan either walks or cycles to school. The probability that he cycles is \(\frac13\).

(a) Write down the probability that he walks. [1]

When he cycles, the probability he is late is \(\frac18\). When he walks, the probability he is late is \(\frac38\).

(b) Complete the tree diagram. [2]

(c(i)) Calculate the probability that Dan cycles and is late. [2]

(c(ii)) Calculate the probability that Dan is not late. [3]

The tree shows the probabilities of a cricket team winning or losing its first two matches.

Find the probability that the team wins at least one match. [3]

Hattie chooses a pen at random from a box. The probability of a red pen is 0.4.

(a) Find the probability that the pen is not red. [1]

The box contains only blue, red and green pens, with 15 blue and 15 green.

(b) Complete the table. [2]

Kiah throws a coin onto a circular board and scores according to where it lands.

A coin touching a line or missing the board scores 0 points. The one-throw probabilities are:

(a) Find \(x\). [2]

(b) For 50 throws, find the expected number scoring 5 points. [1]

For two throws, calculate the probability that:

(c(i)) the first throw scores either 5 or 0; [2]

(c(ii)) the first throw scores 0 and the second scores 5; [2]

(c(iii)) the total is 15 points. [3]

(d) For three throws, find the probability of a total of 10 points. [5]

A train calls at A and then B. It is late at A with probability 0.3. If late at A, it is late at B with probability 0.9. If not late at A, it is late at B with probability 0.2.

(a) Complete the tree diagram. [2]

(b(i)) Find the probability of being late at one or both stations. [3]

(b(ii)) For 250 journeys, find the expected number late at one or both stations. [1]

The train continues to C. The probability of being late at all three stations is 0.27.

(c) Describe briefly what this shows. [1]

A plant produces flowers with probability \(7/8\). Given that it flowers, the flowers are red with probability \(3/4\) and otherwise yellow.

(a(i)) Complete the tree with a probability on each branch. [2]

(a(ii)) Find the probability of red flowers. [2]

(a(iii)) Two plants are chosen at random. Find the probability that both produce red flowers. [2]

(b) Alphonse buys 200 plants. Find the expected number that flower. [2]

(c) Gabriel has 1575 plants with red flowers. Estimate the total number of plants. [2]

Sandra spins a fair eight-sided spinner twice, recording each number. The sectors are labelled 3, 4, 4, 4, 5, 5, 6 and 8.

Find the probability that:

(a) both numbers are 8; [2]

(b) the two numbers are not both 8; [1]

(c) one number is odd and one is even; [2]

(d) the total is at least 13; [3]

(e) the second number is bigger than the first. [3]

Navpreet has 15 cards: 5 squares, 6 triangles and 4 circles. Give probabilities as fractions.

(a) Find the probability a randomly selected card is a circle. [1]

(b) She selects and replaces a card 300 times. Find the expected number of circles. [1]

She selects two cards with replacement. Find the probability that:

(c(i)) one is a square and the other a circle; [3]

(c(ii)) neither is a circle. [3]

She selects two cards without replacement. Find the probability that:

(d(i)) exactly one is a triangle; [3]

(d(ii)) the two shapes differ. [4]

Paul’s probability of winning a race is \(9/35\). Sammy’s is 26%. Who is more likely to win? Give a reason. [2]

Complete the probability table for a biased four-sided die. [2]

Gareth has 8 sweets: 4 orange, 3 lemon and 1 strawberry. He chooses sweets at random without replacement.

(a) He chooses two sweets. Find the probability they have different flavours. [4]

(b) He now chooses a third sweet. Find the probability none of the three is lemon. [2]

Seven cards show the letters: A, A, A, A, B, B, C. Cards are chosen at random.

One card is chosen. Find the probability it:

(a(i)) shows A; [1]

(a(ii)) shows A or B; [1]

(a(iii)) does not show B. [1]

Two cards are chosen without replacement. Find the probability:

(b(i)) both show A; [2]

(b(ii)) their letters are different. [3]

(c) Three cards are chosen without replacement. Find the probability none shows C. [2]

The probability of rain on any day is \(1/5\). Estimate the number of rainy days in a 30-day month. [1]

Two people are chosen at random. Use the table to find the probability they:

(a) both have blue eyes; [2]

(b) have different eye colours. [4]

A box contains 6 red and 8 blue pencils. Two pencils are selected at random without replacement.

(a) Complete the tree diagram. [2]

Find the probability:

(b(i)) both are red; [2]

(b(ii)) at least one is red. [3]

Samira takes part in two charity runs. She has probability 0.8 of finishing each, as shown.

Find the probability she finishes at least one run. [3]

A biased spinner has probabilities:

(a(i)) Find P(2 or 3). [2]

(a(ii)) How many scores of 4 are expected in 5000 spins? [1]

(a(iii)) Find P(1 on the first spin and 4 on the second). [2]

(b) A bag holds 7 red and 5 blue discs. Two are drawn without replacement. Find P(at least one red), as a fraction. [3]

P(fine weather) = ¾. Carlos is late with probability 1/10 if fine, and ⅓ otherwise.

(a) Complete a tree: fine / not fine, then late / not late for each weather branch. [3]

(b) How many fine days are expected in a 60-day term? [1]

(c) Find P(fine and late). [2]

(d) Find P(not late). [3]

(e) Find P(at least one day not fine in a five-day week). [2]

Six cards bear the block capital letters N A T I O N. Give all answers as fractions.

(a(i)) A random card has T printed on it. Find this probability. [1]

(a(ii)) Find the probability it does not have N. [1]

(a(iii)) Find the probability its letter has no lines of symmetry. [1]

(b) Lara chooses a card, replaces it and chooses again. Find the probability that exactly one chosen card has N. [3]

(c) Jacob chooses without replacement until he chooses N. Find the probability that the first N is his fourth card. [3]

After a rainy day, P(rain next day) = 0.4; after a dry day it is 0.2. It rained on Sunday.

(a) Complete the Monday/Tuesday tree. [2]

(b) Find the probability of rain on at least one of Monday and Tuesday. [3]

Two independent fair six-sided dice are rolled. Their sum is the number of spaces moved on a board.

(a(i)) Find P(move 2), as a simplified fraction. [2]

(a(ii)) Find P(move 10), as a simplified fraction. [3]

(b) Give the most likely number of spaces and explain. [2]

To win, Kenwyn must land exactly on space 100. He starts on 97. If the next throw reaches 99 he moves back to 96. If it passes 100 he stays on 97.

(c) Find the probability of reaching 100 in either of his next two throws. [5]

P(Yeung finishes a triathlon) = 3/5; P(Ariven finishes) = 2/3.

(a(i)) Who is more likely to finish? Give a reason. [1]

(a(ii)) Find P(both finish). [2]

(a(iii)) Find P(exactly one finishes). [3]

Yeung then enters two more races. The tree gives the finish probabilities at each stage.

(b(i)) Complete the tree. [3]

(b(ii)) Find P(Yeung finishes all three). [2]

(b(iii)) Find P(Yeung finishes at least one). [3]

Each of two spinners has equally likely numbers 1 to 5. Both are spun once.

(a) Find the probability both show the same number. [2]

(b) Find the probability their sum is 7. [2]

A hotel has 300 rooms numbered from 100 to 399. One is chosen at random. Find the probability its number ends in zero. [2]

Ivan wears a hat with probability \(5/8\). Given a hat, his scarf probability is \(2/3\); given no hat, it is \(1/6\). Give all answers as fractions.

(a) Complete the probability tree. [3]

(b(i)) Find P(no hat and no scarf). [2]

(b(ii)) Find P(hat but no scarf). [2]

(b(iii)) Find P(hat or scarf, but not both). [2]

(c) Given both hat and scarf, P(gloves) = 7/10. Find the probability he does not wear all three. [3]

A box contains 7 red and 3 blue cards. Two cards are drawn randomly without replacement.

(a) Complete the tree. [3]

(b) Find the probability the cards differ in colour, as a fraction. [3]

There are 50 cars. R = red cars; T = two-door cars.

(a(i)) A random car is red and two-door. Find the probability. [1]

(a(ii)) Find P(not red and two-door). [1]

(b) A two-door car is chosen randomly. Find P(not red). [1]

(c) Two cars are chosen randomly. Find P(both red). [2]

(d) Shade \(R\cup T'\) on the Venn diagram. [1]

One of the six letters S P A C E S is chosen at random.

(a) Find the probability of S. [1]

(b) The letter is replaced and the random choice is repeated 600 times. How often would S be expected? [1]

There are 11 cards, labelled E N L A R G E M E N T. Prettie draws two cards at random without replacement.

(a(i)) Find P(L then G). [2]

(a(ii)) Find P(E twice). [2]

(a(iii)) Find P(two identical letters). [2]

She draws a third card, still without replacement.

(b(i)) Find P(all three identical). [2]

(b(ii)) Find P(no E). [2]

(b(iii)) Find P(exactly two identical letters). [5]

A box holds 3 red, 2 blue and 4 green pencils. Raj takes 2 without replacement. Give fractions.

(a) Find P(both red). [2]

(b) Find P(same colour). [3]

(c) Find P(exactly one green). [3]

Give all probabilities as simplified fractions. P(rain today) = 1/3. If it rains today, P(rain tomorrow) = 2/5; if it does not rain today, P(rain tomorrow) = 1/6.

(a(i)) Find P(no rain today). [1]

(a(ii)) Complete the tree. [2]

(b) Find P(rain on at least one day). [3]

(c) Find P(rain on exactly one day). [3]

The 240 flight passengers are shown. H means on holiday; C means hiring a car.

(a(i)) Find n(H). [1]

(a(ii)) How many hire a car but are not on holiday? [1]

(b) Find \(n(H\cup C^{\prime})\). [1]

(c(i)) Find P(hiring a car). [1]

(c(ii)) Find P(on holiday and hiring a car). [1]

Two passengers are chosen at random without replacement. Give the following to 4 decimal places.

(d(i)) Find P(both on holiday). [2]

(d(ii)) Find P(exactly one on holiday). [3]

(e) Two are now chosen from those on holiday. Find P(both hiring a car), to 4 decimal places. [3]

Five discs are numbered 1, 2, 2, 3, 4. Two are selected at random without replacement.

(a(i)) Find the probability both are numbered 2. [2]

(a(ii)) Find the probability their numbers total 5. [3]

(a(iii)) Find the probability their numbers do not total 5. [1]

For 50 students, E = those with an English parent and F = those with a French parent. \(n(E)=15,\ n(F)=9,\ n((E\cup F)')=33\).

(b(i)) Find \(n(E\cap F)\). [1]

(b(ii)) Find \(n(E'\cup F)\). [1]

(b(iii)) Find the probability a random student has both an English and a French parent. [1]

(b(iv)) A student with a French parent is selected. Find the probability they also have an English parent. [1]

Give all answers as fractions. Monday’s rain probability is 3/5. If Monday is rainy, Tuesday’s rain probability is 4/7; if Monday is dry, Tuesday’s rain probability is 5/7.

(a) Complete the tree diagram. [3]

(b)(i) Find the probability of rain on both days. [2]

(b)(ii) Find the probability of rain on Monday but not Tuesday. [2]

(b)(iii) Find the probability of rain on exactly one of the two days. [2]

(c) After two dry days, the probability Wednesday is dry is 1/4. Find the probability of rain on at least one of the three days. [3]

A plant flowers with probability \(7/10\). Given that it flowers, the probability of red is \(2/3\) and of yellow is \(1/4\). The only other flower colour is orange.

(a) Draw a labelled tree diagram, giving every branch probability. [5]

(b) Find the probability that a plant will not produce a yellow flower. [3]

(c) For 120 plants, find the expected number of orange flowers. [2]

In a flu epidemic, 45% of people have a sore throat. Given a sore throat, the probability of no flu is 0.4. Given no sore throat, the probability of flu is 0.2.

Find the probability that a randomly chosen person has flu. [4]

A bag contains 6 white and 2 blue buttons. Emile takes 3 without replacement.

(a)(i) Find the probability all three are white. [3]

(a)(ii) Find the probability exactly one is blue. [3]

Another bag has 25 buttons, x blue. Two are selected without replacement; the probability both are blue is \(7/100\).

(b)(i) Show \(x^2-x-42=0\). [4]

(b)(ii) Factorise \(x^2-x-42\). [2]

(b)(iii) Solve the quadratic. [1]

(b)(iv) Find the number of buttons that are not blue. [1]

(a)(i) A card is selected from A with replacement. Find the probability of M. [1]

(a)(ii) For 100 repetitions, find the expected number of M cards. [1]

(b) Two cards are selected from A without replacement. Find the probability both are S. [2]

(c) One card is selected from each set. Find the probability exactly one is U. [3]

(d) Cards are taken from B without replacement until I or U appears. Find the probability this first happens on the fourth card. [2]

Box A has 3 black balls and 1 white; Box B has 3 black and 2 white.

(a) Which box gives a greater probability of a white ball? Explain. [1]

Abdul chooses A with probability 2/3, otherwise B, then chooses a random ball.

(b)(i) Complete the four missing branch probabilities. [4]

(b)(ii) Find the probability of A and a black ball. [2]

(b)(iii) Find the probability of a black ball. [2]

(c) Tatiana chooses a box with the same probabilities, then takes two balls without replacement. Find the probability both are white. [2]

Each of the six sections is equally likely. Use the shading in the diagram.

(a)(i) For one spin find P(shaded). [1]

(a)(ii) Find P(number 1). [1]

(a)(iii) Find P(shaded and number 1). [1]

(a)(iv) Find P(shaded or number 1). [1]

(b)(i) The spinner is spun twice. Find P(total 20). [2]

(b)(ii) Find P(total 11). [2]

(c)(i) Given shaded, find P(number 2). [1]

(c)(ii) Given number 2, find P(shaded). [1]

(d) It is spun until the first number 2. The probability that this occurs on spin n is 16/243. Find n. [2]

The arrow is equally likely to stop in any of the 10 sectors.

(a) Complete the probabilities. [1]

(b)(i) For one spin, find the most likely number. [1]

(b)(ii) Find the probability of a number less than 4. [1]

(c)(i) For two spins, find the probability both are 2. [1]

(c)(ii) Find the probability first 3, then 4. [2]

(c)(iii) Find the probability the numbers total 4. [3]

(d) The arrow is spun until the first 4. Find the probability this happens on the third spin. [2]

(a) Complete this table of destinations for 255 students. [3]

(b) Find P(random student is a girl going to Europe). [1]

A bag has 7 red and 4 green sweets. Aimee randomly takes and eats two sweets, without replacement.

(a) Complete the tree probabilities. [3]

(b)(i) Find P(two red). [2]

(b)(ii) Find P(one of each colour). [3]

She takes and eats a third sweet.

(c)(i) Find P(three red). [2]

(c)(ii) Find P(at least one red among the three). [3]

Sacha walks or cycles to school. Each day P(walks) = 3/5.

(a)(i) A term has 55 days. Find the expected number of walking days. [1]

(a)(ii) Find the probability he walks on each of the first five days. [2]

P(late | walks) = 1/4; P(late | cycles) = 1/8.

(b)(i) Complete the four missing probabilities. [3]

(b)(ii) Find P(cycles and late). [2]

(b)(iii) Find P(late). [2]