9.2 Fundamentals

105

sigma summation Underscript j equals 1 Overscript n Endscripts upper P left brace upper A intersection upper E Subscript j Baseline right brace equals sigma summation Underscript j equals 1 Overscript m Endscripts upper P left brace upper E Subscript j Baseline right brace equals upper P left brace upper A right brace period

n

E

j=1

P{A ∩E j} =

m

E

j=1

P{E j} = P{A} .

(9.22)

This result can be used to write the denominator of the right-hand side of Eq. (9.18)

as upper P left brace upper A vertical bar upper E Subscript k Baseline right brace upper P left brace upper E Subscript k Baseline right brace divided by upper P left brace upper A right braceP{A|Ek}P{Ek}/P{A}, but this, according to Eq. (9.16) and after cancelling,

equalsupper P left brace upper A intersection upper E Subscript k Baseline right brace divided by upper P left brace upper A right brace equals upper P left brace upper E Subscript k Baseline intersection upper A right brace divided by upper P left brace upper A right braceP{A ∩Ek}/P{A} = P{Ek ∩A}/P{A}, which, again using Eq. (9.16), equals

upper P left brace upper E Subscript k Baseline vertical bar upper A right braceP{Ek|A}. QED.

9.2.3

Bernoulli Trials

Bernoulli trials are defined as repeated (stochastically) independent trials ⁸ (hence,

probabilities multiply) with only two possible outcomes per trial—success (s) or

failure (f)—with respective constant (throughout the sequence of trials) probabilities

pp and q equals 1 minus pq = 1 −p. The sample space of each trial is StartSet s comma f EndSet{s, f}, and the sample space of

nn trials contains 2 Superscript n2ⁿ points. The event “kk successes, with k equals 0 comma 1 comma period period period comma nk = 0, 1, ..., n, and n minus kn −k

failures innn trials” can occur in as many ways askk letters can be distributed amongnn

places (the order of successes and failures does not matter), and each of theSuperscript n Baseline upper C Subscript k Baseline equals StartBinomialOrMatrix n Choose k EndBinomialOrMatrixⁿCk =

(n

k

)

points has probability p Superscript k Baseline q Superscript n minus kp^kq^n−k. Hence, the probability of exactly kk successes in nn

trials is

b left parenthesis k semicolon n comma p right parenthesis equals StartBinomialOrMatrix n Choose k EndBinomialOrMatrix p Superscript k Baseline q Superscript n minus k Baseline periodb(k; n, p) =

(n

k

)

p^kq^n−k .

(9.24)

This function is known as the binomial distribution because the terms are those of

the expansion of left parenthesis a plus b right parenthesis Superscript n(a + b)ⁿ (cf. Sect. 8.3).

Bernoulli trials are easily generalized to more than two outcomes. If the probability

of realizing an outcome upper E Subscript iEi is p Subscript i Baseline left parenthesis i equals 1 comma 2 comma ellipsis comma r right parenthesispi (i = 1, 2, . . . ,r) subject only to the condition

p 1 plus p 2 plus midline horizontal ellipsis plus p Subscript r Baseline equals 1 commap1 + p2 + · · · + pr = 1 ,

(9.25)

then the probability that in nn trials, upper E 1E1 occurs k 1k1 times, upper E 2E2 occurs k 2k2 times, and so

on is

StartFraction n factorial Over k 1 factorial k 2 factorial midline horizontal ellipsis k Subscript r Baseline factorial EndFraction p 1 Superscript k 1 Baseline p 2 Superscript k 2 Baseline midline horizontal ellipsis p Subscript r Superscript k Super Subscript r Superscript Baseline comma

n!

k1!k2! · · · kr! ^pk1

1 ^pk2

2 ^{· · · pkr}

r ^,

(9.26)

where

k 1 plus k 2 plus midline horizontal ellipsis plus k Subscript r Baseline equals n periodk1 + k2 + · · · + kr = n .

(9.27)

8 Stochastic independence is formally defined via the condition

upper P left brace upper A upper H right brace equals upper P left brace upper A right brace upper P left brace upper H right brace commaP{AH} = P{A}P{H} ,

(9.23)

which must hold if the two events upper AA and upper HH are stochastically (sometimes called statistically)

independent.