Probability review Problem set A

A family has two children.

1. (a) Given that at least one is a girl, find the probability that both are girls.
2. (b) Given that the younger child is a girl, find the probability that both are girls.

A broadband connection fails on any given day with probability 0.08, independently of other days.

1. (a) Find the probability that the connection fails on at least one of 3 days.
2. (b) Find the probability that the connection fails on all of 3 days.

A drawer holds 9 black socks and 6 white socks. Two socks are taken out at random one after the other, without replacement.

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

Of 80 members of an arts club, 30 paint, 34 sculpt and 26 print. 10 paint and sculpt, 9 sculpt and print, 8 paint and print, and 5 do 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 prints but does not paint or sculpt.
3. (c) Given that a person chosen at random prints, find the probability that they also sculpt.

Each of 32 pupils has at least one of a phone (P), a smart watch (W) or a tablet (T). 15 have a phone, 16 have a smart watch and 14 have a tablet. 5 have a phone and a smart watch, 6 have a smart watch and a tablet, and 4 have a phone and a tablet.

1. (a) Let x be the number of pupils who have all three items. Draw a Venn diagram showing this information, with each region in terms of x.
2. (b) Show that \(x+30=32\).
3. (c) Find the probability that a pupil chosen at random has all three.
4. (d) Given that a pupil does not have a tablet, find the probability that they have a phone.
5. (e) Given that a pupil has exactly two of the three, find the probability that they have a smart watch and a tablet.

A family has three children. Write B for a boy and G for a girl, eldest first.

1. (a) List the 8 equally likely outcomes.
2. (b) Given that at least one is a boy, find the probability that all three are boys.
3. (c) Given that the eldest child is a boy, find the probability that all three are boys.

On a school day, Ben is late with probability 0.1 and Cara is late with probability 0.2. They are late independently. Find the probability that at least one of them is late.