Probability: exam review Exam review: second paper

A bag holds 12 identical discs numbered 1 to 12. A disc is taken at random, its number is noted and it is put back. Each draw is independent. [5]

1. (a) Find the probability that a disc shows a number less than 5. [1]
2. (b) Four discs are drawn, one at a time, each put back. Find the probability that at least one disc shows a number less than 5. [2]
3. (c) A disc is drawn and put back 150 times. Find the expected number of times that the disc shows a number less than 5. [1]
4. (d) In a test, a number less than 5 came up on 21 of 45 draws. Write down the relative frequency of a number less than 5 in the test. [1]

A gym has 90 members. 52 members use the pool and 45 members use the sauna. Twelve members use neither. [5]

1. (a) Find the number of members who use both the pool and the sauna. [2]
2. (b) A member is chosen at random. Find the probability that they use the pool only. [1]
3. (c) Find the probability that the member uses exactly one of the pool and the sauna. [2]

A fair spinner with five equal sectors numbered 1 to 5 is spun and a fair six-sided die is rolled. The score is the product of the two numbers. The sample space diagram shows every score. [6]

1. (a) Find the probability that the score is 6. [2]
2. (b) Find the probability that the score is greater than 18. [2]
3. (c) The spinner and die are used 150 times. Find the expected number of times that the score is odd. [2]

The two-way table shows the favourite drink of 132 students. Three of the numbers are missing. [7]

1. (a) Find the three missing numbers. [2]
2. (b) Find the probability that a student chosen at random is in Year 12 and prefers juice. [1]
3. (c) Given that a student chosen at random is in Year 11, find the probability that they prefer juice. [2]
4. (d) Determine whether the events "Year 11" and "prefers tea" are independent. [2]

E and F are events with P(E) = 0.62, P(F) = 0.35 and P(E ∪ F) = 0.85. [5]

1. (a) Find \(P(E\cap F)\). [2]
2. (b) State whether E and F are mutually exclusive. Give a reason. [1]
3. (c) G is an event that is mutually exclusive with E, and \(P(E\cup G) = 0.95\). Find \(P(G)\). [1]
4. (d) Find the probability that neither E nor F occurs. [1]

120 people are asked about pets. 25 own both a cat and a dog. 14 own neither. 68 own a dog. [6]

1. (a) Find the number of people who own a cat but not a dog. [2]
2. (b) A person who owns a cat is chosen. Find the probability that they also own a dog. [2]
3. (c) A person who does not own a dog is chosen. Find the probability that they own a cat. [2]

A cow has four calves, born one at a time. Assume that each calf is equally likely to be a bull or a heifer, independently of the others. [6]

1. (a) Find the probability that exactly 3 of the 4 calves are heifers. [2]
2. (b) Given that at least one is a heifer, find the probability that exactly 3 are heifers. [2]
3. (c) Given that the first calf born is a heifer, find the probability that exactly 3 are heifers. [2]

U and V are events with P(U) = 0.6, P(U ∩ V) = 0.27 and P(U ∪ V) = 0.78. [6]

1. (a) Find \(P(V)\). [2]
2. (b) Find the probability that exactly one of the events U and V occurs. [2]
3. (c) Show that U and V are independent. [2]

A bag holds 5 red counters and 3 blue counters. Two counters are taken at random, one after the other, without replacement. [5]

1. (a) Find the probability that the two counters are the same colour. [3]
2. (b) Given that the two counters are the same colour, find the probability that both are red. [2]

Three fair six-sided dice are thrown at the same time and the three scores are added. [5]

1. (a) Write down the number of equally likely outcomes. [1]
2. (b) Find the number of outcomes for which the sum of the three scores is 7. [2]
3. (c) Find the probability that the sum of the three scores is 7. [1]
4. (d) The three dice are rolled 432 times. Find the expected number of times that the sum is 7. [1]

A drawer holds 8 red socks and 6 grey socks. Two socks are taken at random, one after the other, without replacement. The tree diagram shows the situation. [12]

1. (a) Write down the two probabilities that are missing from the tree diagram. [2]
2. (b) Find the probability that both socks are red. [2]
3. (c) Find the probability that the two socks are different colours. [3]
4. (d) Given that the second sock is grey, find the probability that the first sock is red. [3]
5. (e) A third sock is taken, also without replacement. Find the probability that at least one of the three socks is grey. [2]

Each of 80 students was asked whether they play football (F), hockey (H) and rugby (R). The Venn diagram shows the results, where x is a number. [13]

1. (a) Show that x = 10. [2]
2. (b) Find the probability that a student belongs to exactly one of the three sports. [2]
3. (c) Given that a student belongs to Rugby (R), find the probability that the student also belongs to Football (F). [2]
4. (d) Find \(P(F\cup R)\). [2]
5. (e) Two of the 80 students are chosen at random, without replacement. Find the probability that both belong to Football (F). [3]
6. (f) Given that a student belongs to at least one sport, find the probability that they belong to exactly two. [2]

In a city, R is the event that a commuter uses the rail line and S is the event that a commuter uses the shuttle bus. P(R) = 0.55, P(S | R) = 0.4 and P(R ∪ S) = 0.81. [12]

1. (a) Find \(P(R\cap S)\). [2]
2. (b) Find \(P(S)\). [2]
3. (c) Find \(P(R\mid S)\). [2]
4. (d) State whether R and S are mutually exclusive. Give a reason. [1]
5. (e) Show that R and S are not independent. [2]
6. (f) Four students are chosen at random. Assume that their choices are independent. Find the probability that at least one of them uses rail or the shuttle. [3]

A shop sends each parcel by one of three couriers, chosen at random with equal chances. The probability that a parcel arrives damaged is 0.05 with courier X, 0.03 with courier Y and 0.04 with courier Z. The tree diagram shows the situation. [13]

1. (a) Write down the two probabilities that are missing from the tree diagram. [2]
2. (b) Find the probability that a parcel is sent by courier X and arrives damaged. [1]
3. (c) Find the probability that a parcel arrives damaged. [3]
4. (d) Given that a parcel arrives damaged, find the probability that it was sent by courier X. [3]
5. (e) Four parcels are sent. Assume that the parcels are independent. Find the probability that exactly one of the four parcels arrives damaged. [2]
6. (f) Find the probability that at least one of the four parcels arrives damaged. [2]