Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.2 PROBABILITY

However, we may write Pr(A 1 ∩B)as

Pr(A 1 ∩B)=Pr[A 1 ∩(A 2 ∪A 3 )]

=Pr[(A 1 ∩A 2 )∪(A 1 ∩A 3 )]

=Pr(A 1 ∩A 2 )+Pr(A 1 ∩A 3 )−Pr(A 1 ∩A 2 ∩A 3 ).

Substituting this expression, and that for Pr(B) obtained from (30.9), into (30.14)

we obtain the probability addition law for three general events,

Pr(A 1 ∪A 2 ∪A 3 )=Pr(A 1 )+Pr(A 2 )+Pr(A 3 )−Pr(A 2 ∩A 3 )−Pr(A 1 ∩A 3 )

−Pr(A 1 ∩A 2 )+Pr(A 1 ∩A 2 ∩A 3 ). (30.15)

Calculate the probability of drawing from a pack of cards one that is an ace or is a spade

or shows an even number(2,4,6,8,10).

If, as previously,Ais the event that an ace is drawn, Pr(A)= 524. Similarly the eventB,
that a spade is drawn, has Pr(B)=^1352. The further possibilityC, that the card is even (but
not a picture card) has Pr(C)=^2052. The two-fold intersections have probabilities

Pr(A∩B)=

1

52

, Pr(A∩C)=0, Pr(B∩C)=

5

52

.

There is no three-fold intersection as eventsAandCare mutually exclusive. Hence

Pr(A∪B∪C)=

1

52

[(4+13+20)−(1+0+5)+(0)]=

31

52

.

The reader should identify the 31 cards involved.

When the probabilities are combined to calculate the probability for the union

of thengeneral events, the result, which may be proved by induction uponn(see

the answer to exercise 30.4), is

Pr(A 1 ∪A 2 ∪···∪An)=

∑

i

Pr(Ai)−

∑

i,j

Pr(Ai∩Aj)+

∑

i,j,k

Pr(Ai∩Aj∩Ak)

−···+(−1)n+1Pr(A 1 ∩A 2 ∩···∩An). (30.16)

Each summation runs over all possible sets of subscripts, except those in which

any two subscripts in a set are the same. The number of terms in the summation

of probabilities ofm-fold intersections of thenevents is given bynCm(as discussed

in section 30.1). Equation (30.9) is a special case of (30.16) in whichn= 2 and

only the first two terms on the RHS survive. We now illustrate this result with a

worked example that hasn= 4 and includes a four-fold intersection.