Probability

Exercise 1

The number of boys in a family with $4$ children is a stochastic variable $J$ . Assume that the probability of the birth of a girl is the same as the probability of the birth of a boy.

a

Make a table with the probability distribution of $J$ .

b

What expectation do you have of the number of boys in the family? Use a calculation to check whether you are right.

c

When looking at $150$ of such families (with four children), what is the total number of boys? Explain your answer.

Exercise 2

From a vase containing $30$ red and $3$ green marbles $4$ marbles are taken one by one.

a

$X$ denotes the number of green marbles taken when each marble is replaced before the next one is taken from the vase. Make the probability distribution of $X$ .

b

$Y$ denotes the number of green marbles when the marbles are not replaced. Make the probablility distribution of $Y$ .

Exercise 3

From a class with $16$ girls and $12$ boys four pupils are chosen. $M$ is the number of girls in that group of four.

a

What values can $M$ have?

b

Make the probabilty distribution for $M$ .

c

Determine the expected number of girls in the group of four.

Exercise 4

You are throwing two dice and multiplying the number of pips of the one dice with the number of pips on the other one. That gives you the value of the stochastic variable $Z$ .

a

Make the probability distribution of $Z$ .

b

You receive the value of $Z$ in euro’s. Would you want to play that game for a stake of 12 euros? What is the probability of winning something when you play one game?

Exercise 5

In the men's final of the tennis tournament at Wimbledon the contestants play "best of five": the player who is the first to win $3$ sets becomes the champion. There is a winner after at most $5$ sets, but there can also be a winner after only $3$ sets. When assuming that the players are equal in strength, for both of them the probability to win is $50$ %. The number of sets played is a stochastic variable $S$ .

a

Make a probability distribution for $S$ and calculate the expected number of sets.

b

Assume the Wimbledon tournament has been held $100$ times. What is the expected total numer of sets played in de men's finals?

The real data, however, show something different. See the table covering $90$ finals.

match length	3 sets	4 sets	5 sets
number of times	44	22	24

c

Determine the empirical probability density and expected value of $S$ .

The original assumption was not very good. Now imagine that the probability of winning the first set remains $50$ %, but the probability to win the next set after you won a set is $70$ % (the "winning mood").

d

Make a new probability distribution using this probability (carefully study all different cases).

e

Calculate the expected number of sets using the new probability distribution.

Exercise 6

In a casino you can play the following game. A fair coin is tossed at most $6$ times until "tail" comes up. The player receives 1, 2, 4, 8, 16, or $32$ euros if tail comes up after a toss number of respectively 1, 2, 3, 4, 5, or $6$ times. He does not receive anything when head comes up $6$ times. $Y$ is the payment in euros.

a

Make the probability distribution of $Y$ .

b

What should the stake for this game be so that in the long run the casino does not loose money on this game?

c

What is the probability of receiving at least $16$ euros in this game?

Exercise 7

In a group of $10$ men and $16$ women lots are drawn to assign a role as respresentatives to four of its members. $M$ is the number of assigned men, $V$ the number of assigned women.

a

Calculate $P (M = 0)$ and $P (M = 1)$ (as a percentage, rounded to two decimals).

b

Make a probability distribution for $M$ ( as a percentage, rounded to two decimals).

c

Make a probability distribution for $V$ ( as a percentage, rounded to two decimals).

d

Calculate the expected value of $M$ and $V$ .

Exercises