Each of the four engines on an airplane functions correctly on a given flight with probability .99, and the engines function independently of each other. Assume that the plane can make a safe landing if at least two of its engines are functioning correctly. What is the probability that the engines will allow for a safe landing?

A small boy is lost coming down Mount Washington. The leader of the search team estimates that there is a probability [math]p[/math] that he came down on the east side and a probability [math]1 - p[/math] that he came down on the west side. He has [math]n[/math] people in his search team who will search independently and, if the boy is on the side being searched, each member will find the boy with probability [math]u[/math]. Determine how he should divide the [math]n[/math] people into two groups to search the two sides of the mountain so that he will have the highest probability of finding the boy. How does this depend on [math]u[/math]?

[math]2n[/math] balls are chosen at random from a total of [math]2n[/math] red balls and [math]2n[/math] blue balls. Find a combinatorial expression for the probability that the chosen balls are equally divided in color. Use Stirling's formula to estimate this probability. Using BinomialProbabilities, compare the exact value with Stirling's approximation for [math]n = 20[/math].

Assume that every time you buy a box of Wheaties, you receive one of the pictures of the [math]n[/math] players on the New York Yankees. Over a period of time, you buy [math]m \geq n[/math] boxes of Wheaties.

• Use Theorem to show that the probability that you get all [math]n[/math] pictures is
  [[math]] \begin{eqnarray*} 1 &-& {n \choose 1} \left(\frac{n - 1}n\right)^m + {n \choose 2} \left(\frac{n - 2}n\right)^m - \cdots \\ &+& (-1)^{n - 1} {n \choose {n - 1}}\left(\frac 1n \right)^m. \end{eqnarray*} [[/math]]
  Hint: Let [math]E_k[/math] be the event that you do not get the [math]k[/math]th player's picture.
• Write a computer program to compute this probability. Use this program to find, for given [math]n[/math], the smallest value of [math]m[/math] which will give probability [math]\geq .5[/math] of getting all [math]n[/math] pictures. Consider [math]n = 50[/math], 100, and 150 and show that [math]m = n\log n + n \log 2[/math] is a good estimate for the number of boxes needed. (For a derivation of this estimate, see Feller.^{[Notes 1]})

Notes

Prove the following binomial identity

Hint: Consider an urn with [math]n[/math] red balls and [math]n[/math] blue balls inside. Show that each side of the equation equals the number of ways to choose [math]n[/math] balls from the urn.

Let [math]j[/math] and [math]n[/math] be positive integers, with [math]j \le n[/math]. An experiment consists of choosing, at random, a [math]j[/math]-tuple of positive integers whose sum is at most [math]n[/math].

• Find the size of the sample space. Hint: Consider [math]n[/math] indistinguishable balls placed in a row. Place [math]j[/math] markers between consecutive pairs of balls, with no two markers between the same pair of balls. (We also allow one of the [math]n[/math] markers to be placed at the end of the row of balls.) Show that there is a 1-1 correspondence between the set of possible positions for the markers and the set of [math]j[/math]-tuples whose size we are trying to count.
• Find the probability that the [math]j[/math]-tuple selected contains at least one 1.

Let [math]n\ (\mbox{mod}\ m)[/math] denote the remainder when the integer [math]n[/math] is divided by the integer [math]m[/math]. Write a computer program to compute the numbers [math]{n \choose j}\ (\mbox{mod}\ m)[/math] where [math]{n \choose j}[/math] is a binomial coefficient and [math]m[/math] is an integer. You can do this by using the recursion relations for generating binomial coefficients, doing all the arithmetic using the basic function mod([math]n,m[/math]). Try to write your program to make as large a table as possible. Run your program for the cases [math]m = 2[/math] to 7. Do you see any patterns? In particular, for the case [math]m = 2[/math] and [math]n[/math] a power of 2, verify that all the entries in the [math](n - 1)[/math]st row are 1. (The corresponding binomial numbers are odd.) Use your pictures to explain why this is true.

Lucas^{[Notes 1]} proved the following general result relating to Exercise. If [math]p[/math] is any prime number, then [math]{n \choose j}~ (\mbox{mod\ }p)[/math] can be found as follows: Expand [math]n[/math] and [math]j[/math] in base [math]p[/math] as [math]n = s_0 + s_1p + s_2p^2 + \cdots + s_kp^k[/math] and [math]j = r_0 + r_1p + r_2p^2 + \cdots + r_kp^k[/math], respectively. (Here [math]k[/math] is chosen large enough to represent all numbers from 0 to [math]n[/math] in base [math]p[/math] using [math]k[/math] digits.) Let [math]s = (s_0,s_1,s_2,\dots,s_k)[/math] and [math]r = (r_0,r_1,r_2,\dots,r_k)[/math]. Then

For example, if [math]p = 7[/math], [math]n = 12[/math], and [math]j = 9[/math], then

so that

and this result states that

Since [math]{12 \choose 9} = 220 = 3~(\mbox{mod\ }7)[/math], and [math]{5 \choose 2} = 10 = 3~ (\mbox{mod\ }7)[/math], we see that the result is correct for this example. Show that this result implies that, for [math]p = 2[/math], the [math](p^k - 1)[/math]st row of your triangle in Exercise has no zeros.

Notes

Prove that the probability of exactly [math]n[/math] heads in [math]2n[/math] tosses of a fair coin is given by the product of the odd numbers up to [math]2n - 1[/math] divided by the product of the even numbers up to [math]2n[/math].

Let [math]n[/math] be a positive integer, and assume that [math]j[/math] is a positive integer not exceeding [math]n/2[/math]. Show that in Theorem, if one alternates the multiplications and divisions, then all of the intermediate values in the calculation are integers. Show also that none of these intermediate values exceed the final value.