Probability: exam review Exam review

A game uses a fair spinner with 8 equal sectors numbered 1 to 8. Each spin is independent. [5]

1. (a) Find the probability that a spin lands on a number greater than 5. [1]
2. (b) The spinner is spun three times. Find the probability that at least one spin lands on a number greater than 5. [2]
3. (c) The spinner is spun 200 times. Find the expected number of times that the spinner lands on a number greater than 5. [1]
4. (d) In a test, the spinner landed on a number greater than 5 on 36 of 80 spins. Write down the relative frequency of a number greater than 5 in the test. [1]

In a year group of 60 students, 34 study French and 27 study Spanish. Eight students study neither language. [5]

1. (a) Find the number of students who study both languages. [2]
2. (b) A student is chosen at random. Find the probability that they study French only. [1]
3. (c) Find the probability that the student studies exactly one of the two languages. [2]

A fair four-sided die, numbered 1 to 4, and a fair eight-sided die, numbered 1 to 8, are rolled together. 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 12. [2]
2. (b) Find the probability that the score is greater than 20. [2]
3. (c) The dice are used 160 times. Find the expected number of times that the score is odd. [2]

The two-way table shows how 118 students travel to school. 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 a girl who cycles. [1]
3. (c) Given that a student chosen at random is a boy, find the probability that they cycle. [2]
4. (d) Determine whether the events "boy" and "walks" are independent. [2]

A and B are events with P(A) = 0.55, P(B) = 0.4 and P(A ∪ B) = 0.82. [5]

1. (a) Find \(P(A\cap B)\). [2]
2. (b) State whether A and B are mutually exclusive. Give a reason. [1]
3. (c) C is an event that is mutually exclusive with A, and \(P(A\cup C) = 0.9\). Find \(P(C)\). [1]
4. (d) Find the probability that neither A nor B occurs. [1]

80 people are asked about their transport. 18 own both a bike and a car. 11 own neither. 49 own a bike. [6]

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

A family has three children. Assume that each child is equally likely to be a boy or a girl, independently of the others. [6]

1. (a) Find the probability that exactly 2 of the 3 children are girls. [2]
2. (b) Given that at least one is a girl, find the probability that exactly 2 are girls. [2]
3. (c) Given that the oldest child is a girl, find the probability that exactly 2 are girls. [2]

S and T are events with P(S) = 0.45, P(S ∩ T) = 0.18 and P(S ∪ T) = 0.67. [6]

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

A bag holds 6 green counters and 4 yellow 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 green. [2]

Three fair six-sided dice are rolled together 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 6. [2]
3. (c) Find the probability that the sum of the three scores is 6. [1]
4. (d) The three dice are rolled 540 times. Find the expected number of times that the sum is 6. [1]

A box holds 7 blue pens and 5 black pens. Two pens 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 pens are blue. [2]
3. (c) Find the probability that the two pens are different colours. [3]
4. (d) Given that the second pen is black, find the probability that the first pen is blue. [3]
5. (e) A third pen is taken, also without replacement. Find the probability that at least one of the three pens is black. [2]

Each of 70 students was asked whether they belong to the chess club (C), the drama club (D) and the music club (M). The Venn diagram shows the results, where x is a number. [13]

1. (a) Show that x = 7. [2]
2. (b) Find the probability that a student belongs to exactly one of the three clubs. [2]
3. (c) Given that a student belongs to Music (M), find the probability that the student also belongs to Drama (D). [2]
4. (d) Find \(P(C\cup M)\). [2]
5. (e) Two of the 70 students are chosen at random, without replacement. Find the probability that both belong to Drama (D). [3]
6. (f) Given that a student belongs to at least one club, find the probability that they belong to exactly two. [2]

At a college, M is the event that a student takes music and D is the event that a student takes drama. P(M) = 0.6, P(D | M) = 0.3 and P(M ∪ D) = 0.79. [12]

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

Each day a student travels to school by bus, by bike or on foot. A fair spinner with three equal sectors decides how, so each way is equally likely. The probability that the student is late is 0.3 by bus, 0.1 by bike and 0.2 on foot. 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 the student travels by bus and is late. [1]
3. (c) Find the probability that the student is late. [3]
4. (d) Given that the student is late, find the probability that the student travelled by bus. [3]
5. (e) The student goes to school on three days. Assume that the days are independent. Find the probability that the student is late on exactly one of the three days. [2]
6. (f) Find the probability that the student is late on at least one of the three days. [2]