Advanced book on Mathematics Olympiad

314 6 Combinatorics and Probability

6.3.2 Establishing Relations Among Probabilities

We adopt the usual notation:P (A)is the probability of the eventA,P(A∩B)is the
probability thatAandBoccur simultaneously,P(A∪B)is the probability that either
AorBoccurs,P(A−B)is the probability thatAoccurs but notB, andP(A/B)is the
probability thatAoccurs given thatBalso occurs.
Recall the classical formulas:

• addition formula:

P(A∪B)=P (A)+P(B)−P(A∩B);

• multiplication formula:

P(A∩B)=P (A)P (B/A);

• total probability formula: ifBi∩Bj=∅,i, j= 1 , 2 ,...,n(meaning that they are
  independent), andA⊂B 1 ∪B 2 ∪···∪Bn, then

P (A)=P(A/B 1 )P (B 1 )+P(A/B 2 )P (B 2 )+···+P(A/Bn)P (Bn);

• Bayes’ formula: with the same hypothesis,

P(Bi/A)=

P(A/Bi)P (Bi)

P(A/B 1 )P (B 1 )+P(A/B 2 )P (B 2 )+···+P(A/Bn)P (Bn)

.

In particular, ifB 1 ,B 2 ,...,Bncover the entire probability field, then

P(Bi/A)=

P(Bi)

P (A)

P(A/Bi).

The Bernoulli scheme. As a result of an experiment either the eventAoccurs with
probabilitypor the contrary eventA ̄occurs with probabilityq= 1 −p. We repeat the
experimentntimes. The probability thatAoccurs exactlymtimes is

(n

m

)

pmqn−m. This
is also called the binomial scheme because the generating function of these probabilities
is(q+px)n.
The Poisson scheme. We performnindependent experiments. For eachk,1≤k≤n,
in thekth experiment the eventAcan occur with probabilitypk,orA ̄can occur with
probabilityqk= 1 −pk. The probability thatAoccurs exactlymtimes while then
experiments are performed is the coefficient ofxmin the expansion of

(p 1 x+q 1 )(p 2 x+q 2 )···(pnx+qn).

Here is a problem from the 1970 Romanian Mathematical Olympiad that applies the
Poisson scheme.