The Two Laws of Probability (Bulmer, 1979)

The Law of Addition

If A and B are mutually exclusive events, that is if they cannot both occur together, then the probability that either A or B will occur is equal to the sum of their separate probabilities:

P(A or B) = P(A) + P(B)

In general, if A and B are any two events, not necessarily mutually exclusive, the probability that A or B will occur is the sum of their separate probabilities minus the probability that they both occur:

P(A or B) = P(A) + P(B) - P(A and B)
or

P(A) + P(B) = P(A or B) + P(A and B)

#R-code Table 2(a/b)
> Table2a = rbind(c(359881, 8609), c(340454, 7796))

> rownames(Table2a) = c("Male", "Female")
> Table2a
[,1] [,2]
Male 359881 8609
Female 340454 7796


> colnames(Table2a) = c("Liveborn", "Stillborn")
> Table2a
Liveborn Stillborn
Male 359881 8609
Female 340454 7796


> prop.Table2a = prop.table(Table2a)
> prop.Table2a
Liveborn Stillborn
Male 0.5021082 0.01201133
Female 0.4750035 0.01087703


> sum.prop.Table2a = addmargins(prop.Table2a)
> sum.prop.Table2a
Liveborn Stillborn Sum
Male 0.5021082 0.01201133 0.5141195
Female 0.4750035 0.01087703 0.4858805
Sum 0.9771116 0.02288836 1.0000000


>

If A and B are two events, then the probability that both A and B will occur is equal to the probability that A will occur multiplied by the conditional probability that B will occur given that A has occurred, 

P(A and B) = P(A) x P(B|A)

The conditional probability of an event B givent another event A, written P(B|A), is the probability that B will occur if we consider only those occasions on which A also occurs; it is thus the limiting value of the proportion n(A and B) / n(A), where n(A and B) is the number of times on which both A and B have occurred and n(A) is the number of times on which A has occurred.P(B|A) = n(A and B) / n(A)




When the conditional probability of B given A is equal to the unconditional, or absolute, probability of B the two events are said to be "statistically independent".

If A and B are statistically independent, P(B|A) = P(B) and the law of multiplication takes the simple form

P(A and B) = P(A) x P(B)




The Law of Multiplication