Probability models

Exercise 1

At a neighbourhood party there are three men, four women and five children. A team of four people has to be picked from this group in order to play a game.

a

What is the probability that there are exactly two children in the team?

b

What is the probability that there are at least two women in the team?

c

What is the probability that the team consists of only women and children?

d

How many children to you expect to be in the team?

Exercise 2

A batch of $1000$ canned vegetables has been sitting in storage for a long time. You can assume that the expiration date has been exceeded for 10% of the cans. You randomly choose $8$ cans and check the expiration date. You wonder what the chance would be to find three expired cans in this sample.

a

Is this sampling without replacement?

b

How large is the probability you were wondering about?

c

Now use the binomial model to calculate the same probability. How large is the difference between the two methods?

d

Why is it reasonable to use a binomial model in this case?

e

Determine the probability that you picked no more than $3$ cans where the expiration date has been exceeded?

Exercise 3

The leadership of a political party consists of $20$ people, 40% of which are younger than $28$ years. They use a draw to pick $4$ people to go on a trip abroad.

a

How many people in this group of $4$ do you expect to be younger than $28$ years?

b

Determine the probability that three of the four people are younger than $28$ years.

c

Give an approximation of the same probability using a binomial model. How large is the difference to the actual probability?

At a regional meeting of the same party $100$ members are present. Of these members, 40% are younger than $35$ years. They again use a draw to pick $4$ people to represent the regional chapter at the national party congress.

d

Determine the probability that three of these four representatives are younger than $35$ years.

e

Again give an approximation of this probability using a binomial model. Do you see a large difference again? Explain.

Exercise 4

There are $30$ pralines in a box that all look the same. Only $5$ of these pralines have a cream filling, the rest is filled with caramel. You randomly pick four of the pralines.

a

What is the probability that you pick exactly one praline with a cream filling?

b

What is the probability that you pick two ore more pralines with a cream filling?

c

What is the probability that all but one of the four pralines are filled with cream?

Exercise 5

It is known that 90% of all primary school children are right-handed.
What is the chance that out of a randomly chosen group of $20$ children less than $16$ are right-handed?

Exercises