A fair six-sided die, with faces marked 1, 2, 3, 4, 5, 6, is thrown repeatedly until a 4 is obtained.
(a) Find the probability that obtaining a 4 requires fewer than 6 throws.
On another occasion, the die is thrown 10 times.
(b) Find the probability that a 4 is obtained at least 3 times.
Kayla is competing in a throwing event. A throw is counted as a success if the distance achieved is greater than 30 metres. The probability that Kayla will achieve a success on any throw is 0.25.
(a) Find the probability that Kayla takes more than 6 throws to achieve a success. [2]
(b) Find the probability that, for a random sample of 10 throws, Kayla achieves at least 3 successes. [3]
In a certain large college, 22% of students own a car.
(a) 3 students from the college are chosen at random. Find the probability that all 3 students own a car.
(b) 16 students from the college are chosen at random. Find the probability that the number of these students who own a car is at least 2 and at most 4.
On any given day, the probability that Moena messages her friend Pasha is 0.72.
(a) Find the probability that for a random sample of 12 days Moena messages Pasha on no more than 9 days. [3]
(b) Moena messages Pasha on 1 January. Find the probability that the next day on which she messages Pasha is 5 January. [1]
The company also produces large boxes of sweets. For any large box, the probability that it contains more jellies than chocolates is 0.64. 10 large boxes are chosen at random.
Find the probability that no more than 7 of these boxes contain more jellies than chocolates.
In Greenton, 70% of the adults own a car. A random sample of 8 adults from Greenton is chosen.
Find the probability that the number of adults in this sample who own a car is less than 6.
A competition is taking place between two choirs, the Notes and the Classics. There is a large audience for the competition.
A random sample of 6 people is chosen from the audience.
In Quarendon, 66% of households are satisfied with the speed of their wifi connection.
Find the probability that, out of 10 households chosen at random in Quarendon, at least 8 are satisfied with the speed of their wifi connection.
Hazeem repeatedly throws two ordinary fair 6-sided dice at the same time. On each occasion, the score is the sum of the two numbers that she obtains.
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.
Annan has designed a new logo for a sportswear company. A survey of a large number of customers found that 42% of customers rated the logo as good.
A random sample of 10 customers is chosen. Find the probability that fewer than 8 of them rate the logo as good.
On average, 34% of the people who go to a particular theatre are men.
A random sample of 14 people who go to the theatre is chosen. Find the probability that at most 2 people are men.
The probability that Janice will buy an item online in any week is 0.35. Janice does not buy more than one item online in any week.
Find the probability that, in a 10-week period, Janice buys at most 7 items online.
In a certain country the probability that a child owns a bicycle is 0.65.
A random sample of 15 children from this country is chosen. Find the probability that more than 12 own a bicycle.
The results of a survey by a large supermarket show that 35% of its customers shop online. Six customers are chosen at random. Find the probability that more than three of them shop online.
Jake attempts the crossword puzzle in his daily newspaper every day. The probability that he will complete the puzzle on any given day is 0.75, independently of all other days.
Find the probability that he will complete the puzzle at least three times over a period of five days.
At the Nonland Business College, all students sit an accountancy examination at the end of their first year of study. On average, 80% of the students pass this examination.
A random sample of 9 students who will take this examination is chosen. Find the probability that at most 6 of these students will pass the examination.
In a certain country, 60% of mobile phones sold are made by Company A, 35% are made by Company B and 5% are made by other companies.
Find the probability that, out of a random sample of 13 people who buy a mobile phone, fewer than 11 choose a mobile phone made by Company A.
Vehicles approaching a certain road junction from town A can either turn left, turn right or go straight on. Over time it has been noted that of the vehicles approaching this particular junction from town A, 55% turn left, 15% turn right and 30% go straight on. The direction a vehicle takes at the junction is independent of the direction any other vehicle takes at the junction.
Find the probability that, of the next three vehicles approaching the junction from town A, one goes straight on and the other two either both turn left or both turn right.
In Pelmerdon 22% of families own a dishwasher.
Find the probability that, of 15 families chosen at random from Pelmerdon, between 4 and 6 inclusive own a dishwasher.
The mass of grapes sold on any day is independent of the mass sold on any other day. On 10% of days less than 16 kg of grapes are sold. 12 days are chosen at random. Find the probability that less than 16 kg of grapes are sold on more than 2 of these 12 days.