Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

fighting, kittens are removed at random, one at a time, until peace is restored.

Show, by induction, that the expected number of kittens finally remaining is

N(x, y)=

x

y+1

+

y

x+1

.

30.17 If the scores in a cup football match are equal at the end of the normal period of
play, a ‘penalty shoot-out’ is held in which each side takes up to five shots (from
the penalty spot) alternately, the shoot-out being stopped if one side acquires an
unassailable lead (i.e. has a lead greater than its opponents have shots remaining).
If the scores are still level after the shoot-out a ‘sudden death’ competition takes
place.
In sudden death each side takes one shot and the competition is over if one
side scores and the other does not; if both score, or both fail to score, a further
shot is taken by each side, and so on. Team 1, which takes the first penalty, has
a probabilityp 1 , which is independent of the player involved, of scoring and a
probabilityq 1 (= 1−p 1 ) of missing;p 2 andq 2 are defined likewise.
Define Pr(i:x, y) as the probability that teamihas scoredxgoals aftery
attempts, and letf(M) be the probability that the shoot-out terminates after a
totalofMshots.

(a) Prove that the probability that ‘sudden death’ will be needed is

f(11+) =

∑^5

r=0

(^5 Cr)^2 (p 1 p 2 )r(q 1 q 2 )^5 −r.

(b) Give reasoned arguments (preferably without first looking at the expressions

involved) which show that

f(M=2N)=

(^2) ∑N− 6
r=0

{

p 2 Pr(1 :r, N)Pr(2:5−N+r, N−1)

+q 2 Pr(1 : 6−N+r, N)Pr(2:r, N−1)

}

forN=3, 4 ,5and

f(M=2N+1)=

(^2) ∑N− 5
r=0

{

p 1 Pr(1 : 5−N+r, N)Pr(2:r, N)

+q 1 Pr(1 :r, N)Pr(2:5−N+r, N)

}

forN=3,4.

(c) Give an explicit expression for Pr(i:x, y) and hence show that if the teams

are so well matched thatp 1 =p 2 =1/2then

f(2N)=

(^2) ∑N− 6
r=0

(

1

22 N

)

N!(N−1)!6

r!(N−r)!(6−N+r)!(2N− 6 −r)!

,

f(2N+1)=

(^2) ∑N− 5
r=0

(

1

22 N

)

(N!)^2

r!(N−r)!(5−N+r)!(2N− 5 −r)!

.

(d) Evaluate these expressions to show that, expressingf(M) in units of 2−^8 ,we

have

M 6 7 8 9 10 11+

f(M) 8 24 42 56 63 63

Give a simple explanation of whyf(10) =f(11+).