2 Questions for Chapter 2

2.1 Warm-up

Exercise 2.1 Find the probability that in a family of six children, the second child is a girl but they are not all girls (assuming all outcomes equally likely).

Exercise 2.2 A bag contains three red and two white marbles. These are randomly laid out in a row.

Write down the list of all outcomes, each of which is equally likely. Count by hand the total number of outcomes.
Count by hand the number of outcomes in your list for which the white marbles appear side by side.
From these two counts, derive the probability that the two white marbles appear side by side.

Exercise 2.3 In the following exercises, a deck of cards consists of 52 cards in total—there are 4 suits (hearts, diamonds, clubs, and spades), and 13 denominations: 2, 3, , 10, Jack, Queen, King, and Ace. Every combination of suit and value occurs exactly once, resulting in \(4\times 13=52\) cards in total.

You draw three cards, without replacement, from a well shuffled deck. What is the probability that the first is a King, the second is a Queen, and the last is a Jack, precisely in that order?

Exercise 2.4 An \(n\) card hand is simply a selection of \(n\) out of \(52\) cards from a deck, selected without replacement, whose ordering is irrelevant.

How many three card hands are there?
How many three card hands are there, where each card is a face card (Jack, Queen, or King)?
What is the probability of a three card hand where each card is a face card?

Exercise 2.5 You will need a calculator for these two problems.

In a hand of 13 cards, what is the chance that all cards have different values?
In a hand of 5 cards, what is the chance that all cards have different values?

2.2 Workout

Exercise 2.6 Probability as a branch of mathematics was born in the correspondence between Pascal and Fermat concerning some famous problems which, story has it, were raised by the Chevalier de Méré in 1654.

De Méré knew that it was advantageous to bet on the occurrence of at least one 6 in a series of 4 tosses of a die—maybe this was a common gambler’s experience. He argued that it must be at least as advantageous to bet on the occurrence of at least one pair of 6’s in a series of 24 tosses with a pair of dice. Fortune however disappointed him. He complained to Pascal about “preposterous mathematics which had deceived him”.

Find the probabilities of

at least one 6 in 4 tosses of a die; and
at least one pair of 6’s in 24 tosses of a pair of dice.

Comment on the values you obtain. You will need a calculator.

Exercise 2.7 From a well-shuffled standard pack of 52 cards, you pick two cards, one after the other without replacement. Find the probability that at least one of them is a heart. hint:use either C2 or C6.

Exercise 2.8 A five card hand is called a full house when it contains 3 of one value plus 2 of another value (for instance, 3 sixes and 2 queens).

In how many ways can you pick the suits of the cards that appear in a full house consisting of, say, 3 sixes and 2 queens?

hint: In how many ways can you pick the suits of the 3 sixes? In how many ways can you pick the suits of the 2 queens?
In how many ways can you pick the values that appear in a full house?
How many five card hands are a full house? You may need a calculator.

hint: Use parts a and b.
Find the probability of being dealt a full house in a five card hand. You will need a calculator.

hint: Use part c.

Exercise 2.9 In the national lottery, you identify 6 balls from a set of 49 balls. Then 6 balls are selected at random, without replacement, from the 49 balls and you get a prize if at least 3 of these match the balls you identified, regardless of the order. (We ignore the bonus ball for now.)

What is the probability that you get a prize of some sort? You will need a calculator.

hint: How many outcomes match exactly 3 of the balls you selected? How many outcomes match exactly 4 of the balls you selected?

Exercise 2.10 In the national lottery (see Exercise 2.9), in fact, after the 6 balls are selected at random, a seventh bonus ball is selected at random. You get a very special prize if, of the six balls you identified, exactly 5 were in your selected group of 6, and your remaining selected ball matches the bonus ball.

What is the probability that you get a very special prize? You will need a calculator.

Exercise 2.11 A small village has population 30, of whom 20 support the Red party and 10 support the Blue party. An opinion pollster samples precisely 10 villagers at random. Find the probability that precisely 7 of the 10 villagers in the sample support the Red party. You will need a calculator.

Exercise 2.12 A small village has population 30. 15 villagers are from socio-economic group X (whatever that means), 10 from socio-economic group Y and 5 from socio-economic group Z. There is no overlap between socio-economic groups. Again, 10 villagers are sampled at random. What is the probability that precisely 5 villagers in the sample are from group X, precisely 3 from group Y and precisely 2 from group Z? You will need a calculator.

2.3 Stretch

Exercise 2.13 Following on from Exercise 2.6, De Méré’s other query was the probleme des partis. A and B had contracted to play a set of games each of which has equal chances for each player. The player who first reaches five points would take the stakes. Owing to circumstances beyond their control, they are compelled to stop playing at a point where A has won 3 games and B has won 2 games. How should the stakes be divided?

hint: In the present case with A leading by 3 to 2, at most 4 more games are required before somebody reaches 5 points.

Exercise 2.14 Recall the birthday problem discussed in lectures.

Among 5 people, what is the probability that exactly 2 share a birthday?
Among \(n\) people, what is the probability that exactly \(r\) share a single birthday \((2 \leq r \leq n)\)?
If your answer to part (b) is \(q(n,r)\), explain whether or not \(\sum_{r=2}^n q(n,r)\) is equal to the answer to the birthday problem for \(n\), i.e., the probability that among \(n\) people, at least two share a birthday.

Exercise 2.15 A bag contains 5 red, 3 blue and 3 green beads. You draw out four beads at random i.e. every set of beads is equally likely. What are the chances that you draw:

four beads of one colour;
at least one of each colour;
beads with exactly two of the colours?

hint: For this exercise, counting is easiest if you assume that all beads are distinguishable.

hint: For part c, use parts a and b.

Exercise 2.16 Seven standard cubic dice are thrown.

What is the probability that all six numbers show up on the dice? [You will need a calculator.]
What is the probability that exactly five numbers show up on the dice? [You will need a calculator.]