Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

1.3. The Probability Set Function 21

1.3.2.A random experiment consists of drawing a card from an ordinary deck of
52 playing cards. Let the probability set functionP assign a probability of 521 to
each of the 52 possible outcomes. LetC 1 denote the collection of the 13 hearts and
letC 2 denote the collection of the 4 kings. ComputeP(C 1 ),P(C 2 ),P(C 1 ∩C 2 ),
andP(C 1 ∪C 2 ).

1.3.3. A coin is to be tossed as many times as necessary to turn up one head.
Thus the elements cof the sample spaceC are H, TH, TTH, TTTH,and so
forth. Let the probability set function P assign to these elements the respec-
tive probabilities 21 ,^14 ,^18 , 161 , and so forth. Show thatP(C)=1. LetC 1 ={c:
cisH,TH,TTH,TTTH,orTTTTH}. ComputeP(C 1 ). Next, suppose thatC 2 =
{c:cisTTTTHorTTTTTH}. ComputeP(C 2 ),P(C 1 ∩C 2 ), andP(C 1 ∪C 2 ).

1.3.4.IfthesamplespaceisC=C 1 ∪C 2 and ifP(C 1 )=0.8andP(C 2 )=0.5, find
P(C 1 ∩C 2 ).

1.3.5.Let the sample space beC={c:0<c<∞}.LetC⊂Cbe defined by

C={c:4<c<∞}and takeP(C)=

∫

Ce

−xdx. Show thatP(C) = 1. Evaluate

P(C),P(Cc), andP(C∪Cc).

1.3.6.If the sample space isC={c:−∞<c<∞}and ifC⊂Cis a set for which
the integral

∫

Ce

−|x|dxexists, show that this set function is not a probability set

function. What constant do we multiply the integrand by to make it a probability

set function?

1.3.7.IfC 1 andC 2 are subsets of the sample spaceC, show that

P(C 1 ∩C 2 )≤P(C 1 )≤P(C 1 ∪C 2 )≤P(C 1 )+P(C 2 ).

1.3.8.LetC 1 ,C 2 ,andC 3 be three mutually disjoint subsets of the sample space
C.FindP[(C 1 ∪C 2 )∩C 3 ]andP(Cc 1 ∪Cc 2 ).

1.3.9.Consider Remark 1.3.2.

(a)IfC 1 ,C 2 ,andC 3 are subsets ofC, show that

P(C 1 ∪C 2 ∪C 3 )=P(C 1 )+P(C 2 )+P(C 3 )−P(C 1 ∩C 2 )

−P(C 1 ∩C 3 )−P(C 2 ∩C 3 )+P(C 1 ∩C 2 ∩C 3 ).

(b)Now prove the general inclusion exclusion formula given by the expression

(1.3.13).

Remark 1.3.3.In order to solve Exercises (1.3.10)–(1.3.19), certain reasonable

assumptions must be made.

1.3.10.A bowl contains 16 chips, of which 6 are red, 7 are white, and 3 are blue. If

four chips are taken at random and without replacement, find the probability that:

(a) each of the four chips is red; (b) none of the four chips is red; (c) there is at

least one chip of each color.