Advanced High-School Mathematics

342 CHAPTER 6 Inferential Statistics

xk), where x 1 +x 2 +···+xk =n. A little thought reveals that

these probabilities are given by

P(X 1 =x 1 , X 2 =x 2 , ...,Xk=xk) =

Ñ

n

x 1 ,x 2 ,...,xk

é

px 11 px 22 ···pxkk,

where

Ñ

n

x 1 ,x 2 ,...,xk

é

is themultinomial coefficient

Ñ

n

x 1 ,x 2 ,...,xk

é

=

n!

x 1 !x 2 !···xk!

.

So, suppose that my grading distribution is as follows, and that I

have 20 students. Compute the following probabilities:

(a) P(3 As, 6 Bs, 8 Cs, 2 Ds, and 1 F)

(b) P(3 or 4 As, 5 or 6 Bs, 8 Cs, 2 Ds, and 1 F)

(c) P(everyone passes (D or better)

(d) P(at most 5 people get As)

12. (Gambler’s Ruin) Suppose that we have two players, playerAand
    playerBand that playersAandBhave between themN dollars.
    PlayersAandBnow begin their game where playerAtosses a fair
    coin, winning $1 fromBwhenever she tosses a head and losing and
    losing $1 (and giving it toB) whenever she tosses a tail. Holding
    N fixed, letpi=P(A bankrupts B|A started withidollars). (It
    is clear thatp 0 = 0 and thatpN = 1.)

(a) LetEibe the event thatAbankruptsB, given thatAstarted

withidollars; thenP(Ei) =pi.Now argue that

pi = P(Ei|Awins the first game )P(Awins the first game )

+P(Ei|Bwins the first game )P(Bwins the first game )

=

1

2

pi+1+

1

2

pi− 1.

(b) From the above, obtainpi=ip 1 , i= 1, 2 ,...,N.