Advanced High-School Mathematics

326 CHAPTER 6 Inferential Statistics

Eric, who then tosses the coin. If the result is heads, Eric wins;

otherwise he returns the coin to John. They keep playing until

someone wins by tossing a head.

(a) What is the probability that Eric wins on his first toss?

(b) What is the probability that John wins the game?

(c) What is the proability that Eric wins the game?

8. Letnbe a fixed positive integer. Show that for a randomly-selected
   positive integerx, the probability thatxandnare relatively prime
   is

φ(n)

n

. (Hint: see Exercise 20 on page 64.)
9. Consider the following game. Toss a fair coin, until the first head
is reached. The payoff is simply 2ndollars, wherenis the number
of tosses needed until the first head is reached. Therefore, the
payoffs are

No. of tosses 1 2 3 ··· n ···

Payoff $2 $4 $8 ··· $2n ···

How much would you be willing to pay this game? $10? $20? Ask

a friend; how much would she be willing to play this game? Note

that the expected value of this game is infinite!^7

6.1.3 The random harmonic series (optional discussion)

We close this section with an interesting example from analysis. We

saw on page 265 the harmonic series

∑∞

n=1

1

n

diverges and on page 278

we saw that the alternating harmonic series

∑∞

n=1

(−1)n−^1

n

converges (to

ln 2; see page 302). Suppose now that 1 ,  2 , ...is a random sequence
of +1s and −1s, where we regard eachk as a random variable with
P(k = 1) = P(k =−1) = 1/2. Therefore eachk has mean 0 and

variance 1. What is the probability that

∑∞

n=1

n

n

converges?

(^7) Thus, we have a paradox, often called theSt. Petersburg paradox.