George has a fair five-sided spinner with sides labelled 1, 2, 3, 4, 5. He spins the spinner and notes the number on the side on which the spinner lands.

(a) Find the probability that it takes fewer than 7 spins for George to obtain a 5. [2]

George spins the spinner 10 times.

(b) Find the probability that he obtains a 5 more than 4 times but fewer than 8 times. [3]

Hazeem repeatedly throws two ordinary fair six-sided dice at the same time. On each occasion, the score is the sum of the two numbers that she obtains.

(a) Find the probability that it takes exactly 5 throws of the two dice for Hazeem to obtain a score of 8 or more. [2]

(b) Find the probability that it takes no more than 4 throws of the two dice for Hazeem to obtain a score of 8 or more. [2]

(c) For 8 randomly chosen throws of the two dice, find the probability that Hazeem obtains a score of 8 or more on fewer than 3 occasions. [3]

A children’s wildlife magazine is published every Monday. For the next 12 weeks it will include a model animal as a free gift. There are five different models: tiger, leopard, rhinoceros, elephant and buffalo, each with the same probability of being included in the magazine. Sahim buys one copy of the magazine every Monday.

(a) Find the probability that the first time that the free gift is an elephant is before the 6th Monday. [2]

(b) Find the probability that Sahim will get more than two leopards in the 12 magazines. [3]

(c) Find the probability that after 5 weeks Sahim has exactly one of each animal. [3]

80% of the residents of Kinwawa are in favour of a leisure centre being built in the town. 20 residents of Kinwawa are chosen at random and asked, in turn, whether they are in favour of the leisure centre.

(a) Find the probability that more than 17 of these residents are in favour of the leisure centre. [3]

(b) Find the probability that the 5th person asked is the first person who is not in favour of the leisure centre. [1]

(c) Find the probability that the 7th person asked is the second person who is not in favour of the leisure centre. [2]

Three fair 6-sided dice, each with faces marked 1, 2, 3, 4, 5, 6, are thrown at the same time repeatedly. The score on each throw is the sum of the numbers on the uppermost faces.

(a) Find the probability that a score of 17 or more is first obtained on the 6th throw. [3]

(b) Find the probability that a score of 17 or more is obtained in fewer than 8 throws. [2]

Two fair coins are thrown at the same time. The random variable \(X\) is the number of throws of the two coins required to obtain two tails at the same time.

(a) Find the probability that two tails are obtained for the first time on the 7th throw. [2]

(b) Find the probability that it takes more than 9 throws to obtain two tails for the first time. [2]

Three fair six-sided dice, each with faces marked 1, 2, 3, 4, 5, 6, are thrown at the same time, repeatedly. For a single throw of the three dice, the score is the sum of the numbers on the top faces.

(a) Find the probability that the score is 4 on a single throw of the three dice. [3]

(b) Find the probability that a score of 18 is obtained for the first time on the 5th throw of the three dice. [3]

An ordinary fair die is thrown repeatedly until a 5 is obtained. The number of throws taken is denoted by the random variable \(X\).

(a) Write down the mean of \(X\). [1]

(b) Find the probability that a 5 is first obtained after the 3rd throw but before the 8th throw. [2]

(c) Find the probability that a 5 is first obtained in fewer than 10 throws. [2]

A fair spinner with 5 sides numbered 1, 2, 3, 4, 5 is spun repeatedly. The score on each spin is the number on the side on which the spinner lands.

(a) Find the probability that a score of 3 is obtained for the first time on the 8th spin. [1]

(b) Find the probability that fewer than 6 spins are required to obtain a score of 3 for the first time. [2]

An ordinary fair die is thrown until a 6 is obtained.

(a) Find the probability that obtaining a 6 takes more than 8 throws. [2]

Two ordinary fair dice are thrown together until a pair of 6s is obtained. The number of throws taken is denoted by the random variable \(X\).

(b) Find the expected value of \(X\). [1]

(c) Find the probability that obtaining a pair of 6s takes either 10 or 11 throws. [2]

The score when two fair six-sided dice are thrown is the sum of the two numbers on the upper faces.

(a) Show that the probability that the score is 4 is \(\frac1{12}\). [1]

The two dice are thrown repeatedly until a score of 4 is obtained. The number of throws taken is denoted by the random variable \(X\).

(b) Find the mean of \(X\). [1]

(c) Find the probability that a score of 4 is first obtained on the 6th throw. [1]

(d) Find \(\mathrm P(X<8)\). [2]

Two fair coins are thrown at the same time repeatedly until a pair of heads is obtained. The number of throws taken is denoted by the random variable \(X\).

(a) Find \(E(X)\). [1]

(b) Find the probability that exactly 5 throws are required to obtain a pair of heads. [1]

(c) Find the probability that fewer than 7 throws are required to obtain a pair of heads. [2]

Suri has a bag containing 4 blue discs and 3 yellow discs. She selects one disc at random, notes its colour and returns it to the bag. She repeats this process until she obtains a yellow disc.

(a) Find the probability that Suri obtains a yellow disc for the first time on her 4th selection. [1]

(b) Find the probability that Suri obtains a yellow disc for the second time on her 5th selection. [2]

Tan has a bag containing 4 red discs and 3 green discs. He selects discs at random, one at a time and without replacement, until he obtains a green disc. The random variable \(X\) is the number of discs he needs to select to obtain a green disc.

(c) Draw up the probability distribution table for \(X\). [3]

(d) Find \(\operatorname{Var}(X)\). [3]

Jong has a bag containing 5 red marbles and 7 blue marbles. He chooses marbles from the bag one at a time at random, replacing the marble in the bag each time.

(a) Find the probability that the first red marble that Jong obtains is before his 6th choice. [2]

Kai has a bag containing 5 green marbles and 7 yellow marbles. He chooses marbles one at a time at random, replacing the marble in the bag each time.

(b) Find the probability that the second green marble that Kai obtains is on his 9th choice. [3]

Lam has a bag containing 5 black marbles and 7 white marbles. He chooses 10 marbles one at a time, at random and with replacement.

(c) Find the probability that Lam obtains fewer than 3 black marbles. [3]

Ling has three coins. Each coin is biased so that the probability of obtaining a head is \(\frac13\). Ling throws all three coins at the same time. The random variable \(X\) denotes the number of heads that Ling obtains.

(a) Draw up the probability distribution table for \(X\). [3]

Ling throws all three coins at the same time repeatedly.

(b) Find the probability that he obtains at least two heads for the first time on his 7th throw. [2]

(c) Find the probability that he obtains at least two heads for the second time on his 6th or 7th throw. [3]

A fair 8-sided dice with faces labelled 1, 2, 3, 4, 5, 6, 7, 8 is thrown repeatedly.

(a) Find the probability that a 5 will be obtained for the first time on the 7th throw. [1]

(b) Find the probability that a 5 will be obtained for the first time before the 10th throw. [2]

(c) Find the probability that a 5 will be obtained at least once but no more than three times in 12 randomly chosen throws. [3]

A fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6 is thrown repeatedly until a 6 is obtained.

(a) Find the probability that a 6 is obtained for the first time on the 8th throw. [1]

(b) Find the probability that a 6 is obtained for the third time on the 7th throw. [3]

A coin is biased so that the probability of obtaining a head when it is thrown is 0.4. The coin is thrown repeatedly until the first head is obtained.

(a) Find the probability that the first head is obtained on the 5th throw. [1]

(b) Find the probability that the first head is obtained after the 6th throw. [2]

A fair 6-sided dice with faces labelled 1, 2, 3, 4, 5, 6 is thrown repeatedly. The random variable \(X\) denotes the number of throws required to obtain a 6.

(a) Find \(\operatorname{P}(X=7)\). [1]

(b) Find \(\operatorname{P}(3\le X\le5)\). [2]

(a) Find the number of different arrangements of the 8 letters in the word KANGAROO in which the two As are together and the two Os are not together. [3]

A fair 8-sided dice has faces labelled K, A, N, G, A, R, O, O. The dice is rolled repeatedly.

(b) Find the probability that fewer than 6 rolls of this dice are required to obtain an A. [2]

(c) Find the probability that the second A is obtained on the 6th roll of the dice. [2]

Vehicles approaching a certain road junction from Bromley must go either left, right or straight on. Over time, it is known that 30% turn left, 25% turn right and 45% go straight on. The driver of each vehicle chooses a direction independently of all other drivers.

(a) Find the probability that the next three vehicles approaching this junction from Bromley all go in different directions. [2]

(b) Find the probability that, from the vehicles approaching this junction from Bromley today, the 1st vehicle to go left is before the 9th vehicle. [2]

(c) Find the probability that, from the vehicles approaching this junction from Bromley today, the 2nd vehicle to go left is the 7th vehicle. [2]

A fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6 is thrown repeatedly until a 3 is obtained. The number of throws taken is denoted by the random variable \(X\).

(a) Find \(\operatorname{P}(X=8)\). [1]

(b) Find \(\operatorname{P}(X<9)\). [2]

(c) Find the probability that a 3 is obtained for the second time before the 6th throw. [3]

Nicola throws an ordinary fair six-sided dice. The random variable \(X\) is the number of throws that she takes to obtain a 6.

(a) Find \(P(X<8)\). [2]

(b) Find the probability that Nicola obtains a 6 for the second time on her 8th throw. [2]

Leno independently scores a goal with probability 0.4 on each attempt. \(X\) is the number of attempts until his first goal.

(a) Find \(P(X=5)\). [1]

(b) Find \(P(3\le X\le7)\). [2]

(c) Find the probability that Leno scores his second goal on or before his 5th attempt. [3]

Leno has 75 attempts.

(d) Use a suitable approximation to find the probability that he scores more than 28 but fewer than 35 goals. [5]

Rajesh applies once a year for a music festival ticket. Each application succeeds with probability 0.3, independently of other years.

(a) Find the probability that his first success is on his 7th attempt. [1]

(b) Find the probability that his first success is before his 6th attempt. [2]

(c) Find the probability that his second success is on his 10th attempt. [2]

Sam independently scores a goal on any attempt with probability 0.7.

(a) She makes 10 attempts. Find the probability that she scores on fewer than 8. [3]

(b) Find the probability that her first success is before her 5th attempt. [2]

(c) Wei independently scores with probability 0.6 per attempt and continues until he has scored 3 goals. Find the probability that his third goal is on his 7th attempt. [3]