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Problems

1.1 To Begin or Not to Begin?

An urn contains k black balls and a single red ball. Peter and Paula drawwithout replacement balls from this urn, alternating after each draw until thered ball is drawn. The game is won by the player who happens to draw thesingle red ball. Peter is a gentleman and offers Paula the choice of whethershe wants to start or not. Paula has a hunch that she might be better off ifshe starts; after all, she might succeed in the first draw. On the other hand,if her first draw yields a black ball, then Peter’s chances to draw the red ballin his first draw are increased, because then one black ball is already removedfrom the urn. How should Paula decide in order to maximize her probabilityof winning?

1.2 A Tournament Problem

Ten players participate in the first round of a tennis tournament: 2 femalesand 8 males. Five single matches are fixed at random by successively drawing,without replacement, the names of all 10 players from an urn: the player drawnfirst plays against the one whose name comes up second, the third against thefourth, etc.

a. What is the probability that there will not be a single match involvingtwo female players? Is this probability smaller, equal to, or larger than thecorresponding probability with 20 females and 80 males?

b. Try to answer the general case in which there are 2n players, of whom2 ≤ k ≤ n are female. What is the probability p(k, n) that among the nmatches there will not be a single one involving two female players?

4 Problems

1.3 Mean Waiting Time for 1 − 1 vs. 1 − 2

Peter and Paula play a simple game of dice, as follows. Peter keeps throw-ing the (unbiased) die until he obtains the sequence 1 − 1 in two successivethrows. For Paula, the rules are similar, but she throws the die until sheobtains the sequence 1 − 2 in two successive throws.

a. On average, will both have to throw the die the same number of times?If not, whose expected waiting time is shorter (no explicit calculations arerequired)?

b. Derive the actual expected waiting times for Peter and Paula.

1.4 How to Divide up Gains in Interrupted Games

Peter and Paula play a game of chance that consists of several rounds.Each individual round is won, with equal probabilities of 1

2 , by either Peter orPaula; the winner then receives one point. Successive rounds are independent.Each has staked $50 for a total of $100, and they agree that the game ends assoon as one of them has won a total of 5 points; this player then receives the$100. After they have completed four rounds, of which Peter has won threeand Paula only one, a fire breaks out so that they cannot continue their game.

a. How should the $100 be divided between Peter and Paula?b. How should the $100 be divided in the general case, when Peter needs

to win a more rounds and Paula needs to win b more rounds?

1.5 How Often Do Head and Tail Occur Equally Often?

According to many people’s intuition, when two events, such as head andtail in coin tossing, are equally likely then the probability that these eventswill occur equally often increases with the number of trials. This expectationreflects the intuitive notion that in the long run, asymmetries of the frequen-cies of head and tail will “balance out” and cancel.

To find the basis of this intuition, consider that 2n fair and independentcoins are thrown at a time.

a. What is the probability of an even n : n split for head and tail when2n = 20?

b. Consider the same question for 2n = 200 and 2n = 2000.

Problems 5

1.6 Sample Size vs. Signal Strength

An urn contains six balls — three red and three blue. One of these balls— let us call it ball A — is selected at random and permanently removedfrom the urn without the color of this ball being shown to an observer. Thisobserver may now draw successively — at random and with replacement —a number of individual balls (one at a time) from among the five remainingballs, so as to form a noisy impression about the ratio of red vs. blue ballsthat remained in the urn after A was removed.

Peter may draw a ball six times, and each time the ball he draws turnsout to be red. Paula may draw a ball 600 times; 303 times she draws a redball, and 297 times a blue ball. Clearly, both will tend to predict that ball Awas probably blue. Which of them — if either — has the stronger empiricalevidence for his/her prediction?

1.7 Birthday Holidays

The following problem is described in Cacoullos (1989, pp. 35–36).A worker’s legal code specifies as a holiday any day during which at least

one worker in a certain factory has a birthday. All other days are workingdays. How many workers (n) must the factory employ so that the expectednumber of working man-days is maximized during the year?

1.8 Random Areas

Peter and Paula both want to cut out a rectangular piece of paper. Becausethey are both probabilists they determine the exact form of the rectangle byusing realizations of a positive rv, say U, as follows. Peter is lazy and generatesjust a single realization of this rv; he then cuts out a square that has length andwidth equal to this value. Paula likes diversity and generates two independentrealizations of U. She then cuts out a rectangle with width equal to the firstrealization and length equal to the second realization.

a. Will the areas cut out by Peter and Paula differ in expectation?b. If they do, is Peter’s or Paula’s rectangle expected to be larger?