Probability

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Exercises

Exercise 1

Two tasks need to be done. From a group of three men and five women two people will be drawn by lots who will carry out the tasks. A vase is filled with $8$ marbles, marked m1, m2, m3 and v1, v2, v3, v4, v5. Two marbles are drawn randomly from the vase, without replacement.

What is the probability that the first task is carried out by a man?

Somebody says: "The probability that the second task is carried out by a man is equal to the probability that the first task is carried out by a man, because you might as well draw lots for the second task." Use a calculation to find out whether this is correct.

The tasks are cooking and washing dishes. What is the probability that both tasks will be carried out by a woman?

Now asssume that the first marble is replaced. This may mean that one person needs to do both tasks.

What is the probability that both tasks are carried out by the same person?

What is the probability that both tasks are carried out by one man?

Is it true that the probability that both tasks have to be done by a woman is larger now than at c?

Exercise 2

A bookmaker has established that at Arsenal's home match against Juventus 50% of the visitors thinks that Arsenal will win and one third that Juventus will win. At the return match he estimates it to be different, namely that one third counts on Arsenal to win and one third on Juventus.

Make a probability tree for both matches.

What is the probability that each team wins one match?

Exercise 3

A vase contains 10 marbles, 6 made of wood and $4$ made of plastic. Out of the wooden marbles $4$ are red and $2$ green. Out of the plastic marbles $3$ are red and $1$ is green. You cannot distinguish wood and plastic by touch. Two marbles are drawn from the vase. You note the colour and the material of the marble. Assume that the marble that is drawn first is replaced before the second marble is drawn.

Calculate the probability that a red wooden marble is drawn first, then a green plastic one.

Calculate the probability that you

draw a red wooden marble and a green plastic one.

Now assume that the first marble is not replaced.

Calculate the probability that a red wooden marble is drawn first, then a green plastic one.

Calculate the probability that you draw a red wooden marble and a green plastic one.

If only the colour of the marbles matters you can use a smaller probability tree.

Draw that probability tree for the situation with and without replacement.

For both situations: calculate the probablility of two differently coloured marbles.

That probability is larger if you you don't replace. Explain why.

Exercise 4

Three dice are thrown. The probability tree could now become very large. Maybe you only need a part of it? Or can you imagine a vase?

What is the probability of throwing $17$ or $18$ ?

What is the probability of throwing $16$ ?

What is the probability of throwing at least two sixes?

For the question about sixes you can use a vase model. How many colours do you use? How many marbles of each colour do you need?

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