In Example find the probability that the coin turns up heads for the first time on the tenth, eleventh, or twelfth toss.

A die is rolled until the first time that a six turns up. We shall see that the probability that this occurs on the [math]n[/math]th roll is [math](5/6)^{n-1}\cdot(1/6)[/math]. Using this fact, describe the appropriate infinite sample space and distribution function for the experiment of rolling a die until a six turns up for the first time. Verify that for your distribution function [math]\sum_{\omega} m(\omega) = 1[/math].

Let [math]\Omega[/math] be the sample space

and define a distribution function by

for some fixed [math]r[/math], [math]0 \lt r \lt 1[/math], and for [math]j = 0, 1, 2, \ldots[/math]. Show that this is a distribution function for [math]\Omega[/math].

Our calendar has a 400-year cycle. B. H. Brown noticed that the

number of times the thirteenth of the month falls on each of the days of the week in the 4800 months of a cycle is as follows:

From this he deduced that the thirteenth was more likely to fall on Friday than on any other day. Explain what he meant by this.

Tversky and Kahneman^{[Notes 1]} asked a group of subjects to carry out the following task. They are told that:

Linda is 31, single, outspoken, and very bright. She majored in philosophy in college. As a student, she was deeply concerned with racial discrimination and other social issues, and participated in anti-nuclear demonstrations.

The subjects are then asked to rank the likelihood of various alternatives, such as:

1. Linda is active in the feminist movement.
2. Linda is a bank teller.
3. Linda is a bank teller and active in the feminist movement.

Tversky and Kahneman found that between 85 and 90 percent of the subjects rated alternative (1) most likely, but alternative (3) more likely than alternative (2). Is it? They call this phenomenon the conjunction fallacy, and note that it appears to be unaffected by prior training in probability or statistics. Is this phenomenon a fallacy? If so, why? Can you give a possible explanation for the subjects' choices?

Notes

Two cards are drawn successively from a deck of 52 cards. Find the probability that the second card is higher in rank than the first card. Hint: Show that [math]1 = P(\mbox{higher}) + P(\mbox{lower}) + P(\mbox{same})[/math] and use the fact that [math]P(\mbox{higher}) = P(\mbox{lower})[/math].

A life table is a table that lists for a given number of births the estimated number of people who will live to a given age. In Appendix C we give a life table based upon 100,00 births for ages from 0 to 85, both for women and for men. Show how from this table you can estimate the probability [math]m(x)[/math] that a person born in 1981 would live to age [math]x[/math]. Write a program to plot [math]m(x)[/math] both for men and for women, and comment on the differences that you see in the two cases.

Here is an attempt to get around the fact that we cannot choose a “random integer.”

• What, intuitively, is the probability that a “randomly chosen” positive integer is a multiple of 3?
• Let [math]P_3(N)[/math] be the probability that an integer, chosen at random between 1 and [math]N[/math], is a multiple of 3 (since the sample space is finite, this is a legitimate probability). Show that the limit
  [[math]] P_3 = \lim_{N \to \infty} P_3(N) [[/math]]
  exists and equals 1/3. This formalizes the intuition in (a), and gives us a way to assign “probabilities” to certain events that are infinite subsets of the positive integers.
• If [math]A[/math] is any set of positive integers, let [math]A(N)[/math] mean the number of elements of [math]A[/math] which are less than or equal to [math]N[/math]. Then define the “probability” of [math]A[/math] as
  [[math]] P(A) = \lim_{N \to \infty} A(N)/N\ , [[/math]]
  provided this limit exists. Show that this definition would assign probability 0 to any finite set and probability 1 to the set of all positive integers. Thus, the probability of the set of all integers is not the sum of the probabilities of the individual integers in this set. This means that the definition of probability given here is not a completely satisfactory definition.
• Let [math]A[/math] be the set of all positive integers with an odd number of digits. Show that [math]P(A)[/math] does not exist. This shows that under the above definition of probability, not all sets have probabilities.

(from Sholander^{[Notes 1]}) In a standard clover-leaf interchange, there are four ramps for making right-hand turns, and inside these four ramps, there are four more ramps for making left-hand turns. Your car approaches the interchange from the south. A mechanism has been installed so that at each point where there exists a choice of directions, the car turns to the right with fixed probability [math]r[/math].

• If [math]r = 1/2[/math], what is your chance of emerging from the interchange going west?
• Find the value of [math]r[/math] that maximizes your chance of a westward departure from the interchange.

Notes

(from Benkoski^{[Notes 1]}) Consider a “pure” cloverleaf interchange in which there are no ramps for right-hand turns, but only the two intersecting straight highways with cloverleaves for left-hand turns. (Thus, to turn right in such an interchange, one must make three left-hand turns.) As in the preceding problem, your car approaches the interchange from the south. What is the value of [math]r[/math] that maximizes your chances of an eastward departure from the interchange?

Notes