Probability models

Exercise 1

You roll two four-sided dice. $X$ is the sum of pips on the sides that end up at the bottom (facing the table).

a

Draw up a probability distribution for $X$ .

b

If you rolled the dice a large number of times, what is the average number of pips you expect to see?

Exercise 2

Ernie has three cards. One card is red on both sides, one card is white on both sides, and the third card has one red side and one white side. Ernie puts all three cards into a top hat and shakes it. Without looking, he draws a card and puts it flat on the table. The side of the card facing up is white. He asks Bert: " Do you want to bet a toy car that the other side of this card is also white?"
Bert accepts the bet. He thinks that his chance of winning is equal to that of Ernie, because the card on the table has to be either white-white or white-red. Bert loses the game. The card was white on the other side.
But Bert does not give up that easily! They play the same game over and over again. Ernie keeps betting that the hidden side of the card is the same color as the side facing up. Bert claims the opposite.

a

Explain why Bert's chance of winning the game is only $\frac{1}{3}$ .

Now that Bert has figured out why he loses so often, Ernie has to look for another victim. Cookie Monster is willing to play the game. Ernie does exactly what he did with Bert. They always play a series of three bets and each of them starts with three cookies. The winner of a bet gets a cookie from the loser. As soon as one player has run out of cookies the game is over.

b

What is the probability that the game is over after one series of three bets?

c

What is the probability that the game is over after four bets?

Exercise 3

The average temperature of a calendar year, the so-called annual temperature, can be a little unpredictable. The annual temperature in 1989, for example, was all of $1.4 ° C$ higher than the average of all annual temperatures from 1900 until 1989.
In the above histogram (with a class width of $0.2 ° C$ ) you can see the frequency distribution of annual temperatures between 1900 and 1989.

A climatologist assumes that the annual temperature is distributed approximately normally over a long period of time. The $90$ annual temperatures have a mean of $9.2 ° C$ and a standard deviation of $0.6 ° C$ .
If you assume a normal distribution of the annual temperatures, then the histogram would appear to noticeably deviate from the bell shape. Of the $90$ years there are $13$ in which the temperature was more than $0.7 ° C$ lower than the mean.

a

How many years would you have expected to have an annual temperature that was more than $0.7 ° C$ lower than the mean? Support your answer with a calculation.

b

Years with an annual temperature that is more than $1.1 ° C$ higher than the mean are called exceptionally warm. Show that you might expect three of such "exceptionally warm" years per century.

Exercise 4

In the card game bridge, $52$ cards are randomly distributed between four people. In order to make good calls you need a simple way to evaluate the cards in your hand. You do this with a point system. An ace is worth $4$ points, a king 3, a queen $2$ and a jack $1$ point. All other cards are not worth any points.
What is the probability that a player has exactly $4$ points in his hand after four cards have been dealt?

Exercise 5

For an experiment you have a pool of $5$ female and $5$ male subjects. They are randomly assigned to two groups A and B of five people each.

a

What is the probability that there are at least $4$ women in group A?

When the experiment is repeated, two groups of five are picked from a pool of $5000$ male and $5000$ female volunteers.

b

What is the chance now that there are at least $4$ women in group A?

Exercise 6

Fizzy drink is dispensed into $0.25$ litre bottles by a machine. The filling volume is normally distributed with a mean of $0.27$ litres and a standard deviation of $0.01$ litres.

a

What is the probability that any one of these bottles contains too little fizzy drink?

b

You buy a crate of $24$ of these bottles. What is the probability that no more than $2$ of the bottles contain too little fizzy drink?

c

The company wants to make sure that no more than 1% of the bottles end up with too little fizzy drink in them. The precision of the bottling machine cannot be improved. Therefore the filling volume will have to be increased. How large will this average volume need to be?

Exercises