9.2 Fundamentals

103

StartLayout 1st Row 1st Column upper P Subscript left bracket 0 right bracket 2nd Column equals 3rd Column 1 minus upper P 1 equals 1 minus 1 plus StartFraction 1 Over 2 factorial EndFraction minus StartFraction 1 Over 3 factorial EndFraction plus minus midline horizontal ellipsis plus or minus StartFraction 1 Over left parenthesis upper N minus 2 right parenthesis factorial EndFraction minus or plus StartFraction 1 Over left parenthesis upper N minus 1 right parenthesis factorial EndFraction plus or minus StartFraction 1 Over upper N factorial EndFraction 2nd Row 1st Column upper P Subscript left bracket 1 right bracket 2nd Column equals 3rd Column 1 minus 1 plus StartFraction 1 Over 2 factorial EndFraction minus StartFraction 1 Over 3 factorial EndFraction plus minus midline horizontal ellipsis plus or minus StartFraction 1 Over left parenthesis upper N minus 2 right parenthesis factorial EndFraction minus or plus StartFraction 1 Over left parenthesis upper N minus 1 right parenthesis factorial EndFraction 3rd Row 1st Column upper P Subscript left bracket 2 right bracket 2nd Column equals 3rd Column StartFraction 1 Over 2 factorial EndFraction left bracket 1 minus 1 plus StartFraction 1 Over 2 factorial EndFraction minus StartFraction 1 Over 3 factorial EndFraction plus minus midline horizontal ellipsis plus or minus StartFraction 1 Over left parenthesis upper N minus 3 right parenthesis factorial EndFraction minus or plus StartFraction 1 Over left parenthesis upper N minus 2 right parenthesis EndFraction factorial right bracket 4th Row 1st Column Blank 2nd Column vertical ellipsis 3rd Column Blank 5th Row 1st Column upper P Subscript left bracket upper N minus 1 right bracket 2nd Column equals 3rd Column StartFraction 1 Over left parenthesis upper N minus 1 right parenthesis factorial EndFraction left brace 1 minus 1 right brace equals 0 6th Row 1st Column upper P Subscript left bracket upper N right bracket 2nd Column equals 3rd Column StartFraction 1 Over upper N factorial EndFraction period EndLayoutP[0] = 1 −P1 = 1 −1 + ¹

2! ⁻¹

3! ^{+ −· · · ±}

1

(N −2)! ^∓

1

(N −1)! ^{± 1}

N!

P[1] = 1 −1 + ¹

2! ⁻¹

3! ^{+ −· · · ±}

1

(N −2)! ^∓

1

(N −1)!

P[2] = ¹

2!

|

1 −1 + ¹

2! ⁻¹

3! ^{+ −· · · ±}

1

(N −3)! ^∓

1

(N −2)^!

|

...

P[N−1] =

1

(N −1)!^{{1 −1} = 0}

P[N] = ¹

N! ^.

Noticing again the similarity with the expansion of 1 divided by e1/e, for large upper NN,

upper P Subscript left bracket m right bracket Baseline almost equals StartFraction e Superscript negative 1 Baseline Over m factorial EndFractionP[m] ≈^e−1

m!

(9.11)

(i.e., a special case of the Poisson distribution withlamda equals 1λ = 1). The probabilityupper P Subscript mPm that

mm or more of the events upper A 1 comma ellipsis comma upper A Subscript upper N BaselineA1, . . . , AN occur simultaneously is

upper P Subscript m Baseline equals upper P Subscript left bracket m right bracket Baseline plus upper P Subscript left bracket m plus 1 right bracket Baseline plus midline horizontal ellipsis plus upper P Subscript left bracket upper N right bracket Baseline periodPm = P[m] + P[m+1] + · · · + P[N] .

(9.12)

Starting with Eq. (9.9) and noting that

upper P Subscript left bracket m plus 1 right bracket Baseline equals upper P Subscript m Baseline minus upper P Subscript left bracket m right bracket Baseline commaP[m+1] = Pm −P[m] ,

(9.13)

by induction, for m greater than or equals 1m ≥1,

StartLayout 1st Row 1st Column upper P Subscript left bracket m right bracket Baseline equals upper S Subscript m Baseline minus StartBinomialOrMatrix m Choose m minus 1 EndBinomialOrMatrix upper S Subscript m plus 1 Baseline 2nd Column plus StartBinomialOrMatrix m plus 1 Choose m minus 1 EndBinomialOrMatrix upper S Subscript m plus 2 2nd Row 1st Column Blank 2nd Column minus StartBinomialOrMatrix m plus 2 Choose m minus 1 EndBinomialOrMatrix upper S Subscript m plus 3 Baseline plus midline horizontal ellipsis plus or minus StartBinomialOrMatrix upper N minus 1 Choose m minus 1 EndBinomialOrMatrix upper S Subscript upper N Baseline period EndLayoutP[m] = Sm −

(

m

m −1

)

Sm+1 +

(m + 1

m −1

)

Sm+2

−

(m + 2

m −1

)

Sm+3 + · · · ±

(N −1

m −1

)

SN .

(9.14)

9.2.2

Conditional Probability

The notion of conditional probability is of great importance. ⁷ It refers to questions

of the type “what is the probability of event upper AA, given that upper HH has occurred?” We use

7 Indeed, Reichenbach, Popper, and others have taken the view that conditional probability may and

should be chosen as the basic concept of probability theory. We should in any case note that most

of the results derived for unconditional probabilities are also valid for conditional probabilities.