Probability review Problem set B

Two fair coins are tossed one after the other.

1. (a) Given that at least one is a head, find the probability that both are heads.
2. (b) Given that the first coin is a head, find the probability that both are heads.

A school bus is cancelled on any given day with probability 0.06, independently of other days.

1. (a) Find the probability that the bus is cancelled on at least one of 3 days.
2. (b) Find the probability that the bus is cancelled on all of 3 days.

A raffle bag holds 10 red tickets and 4 blue tickets. Two tickets are drawn at random one after the other, without replacement.

1. (a) Given that the first is red, find the probability that the second is also red.
2. (b) Find the probability that both are red.
3. (c) Is it more likely that the two are the same colour or different colours? Justify your answer.

Of 90 visitors to a town, 36 went to the museum, 40 to the zoo and 30 to the park. 12 went to the museum and the zoo, 10 to the zoo and the park, 9 to the museum and the park, and 4 went to all three.

1. (a) Draw a Venn diagram and fill in every region, including the region outside the circles.
2. (b) Find the probability that a person chosen at random went to the park only.
3. (c) Given that a person chosen at random went to the park, find the probability that they also went to the zoo.

Each of 35 students takes at least one of art (A), music (M) or drama (D). 16 take art, 19 take music and 17 take drama. 5 take art and music, 8 take music and drama, and 7 take art and drama.

1. (a) Let x be the number of students who take all three subjects. Draw a Venn diagram showing this information, with each region in terms of x.
2. (b) Show that \(x+32=35\).
3. (c) Find the probability that a student chosen at random has all three.
4. (d) Given that a student does not take drama, find the probability that they take art.
5. (e) Given that a student has exactly two of the three, find the probability that they take music and drama.

Three fair coins are tossed one after another. Write H for a head and T for a tail.

1. (a) List the 8 equally likely outcomes.
2. (b) Given that at least one is a head, find the probability that all three are heads.
3. (c) Given that the first coin is a head, find the probability that all three are heads.

Two machines in a factory break down on a given day with probabilities 0.15 and 0.25, independently. Find the probability that at least one of them breaks down on a given day.