Surfstat.australia: an online text in introductory Statistics

VARIATION AND PROBABILITY

CONDITIONAL PROBABILITY

The probability of an event A may have to be recalculated if we know for certain that another event B has already occurred and A and B are not independent.

Example - A family of 3 children

In a family of 3 children suppose you are told that there are fewer than 2 boys. What is the probability that all 3 children are of the same sex?

Using the previous notation

C : all children of the same sex

D : fewer than 2 boys.

We want the probability of C given that D has occurred. We will use the notation P(C|D) to describe this.
Each column lists all outcomes.
Those comprising the events
C and D are in boldface. 		'C' 'D'
GGG GGG
GGB GGB
GBG GBG
GBB GBB
BGG BGG
BGB BGB
BBG BBG
BBB BBB

As D has occurred, only 4 outcomes are now possible: GGG, GGB, GBG and BGG. Their probabilities must be made to sum to 1. To achieve this the probabilities calculated previously need to be "rescaled" by dividing by their total, which was P(D) = 0.47.

The probability of C, given that D has occurred, is called the conditional probability and is written as P(C|D). Recall that the probability of GGG was 0.11:

In general for events X and Y the conditional probability of X given that Y has occurred is

This can also be rearranged to give the useful formulas

P(X and Y) = P(X|Y)P(Y)
P(X and Y) = P(Y|X) P(X)

Example - Gender of employees

The table below shows the probabilities of males (M) and females (F) being employed (E) or unemployed (U) in some population (it excludes those not wishing to be employed).

M F
E 0.52 0.41 0.93
U 0.05 0.02 0.07
0.57 0.43 1.00

Find

(a) P(E|M), the conditional probability of employment given that the person is male
(b) P(M|E), the conditional probability of being male given that the person is employed.

Answers:

Figure 3: Tree model showing conditional probabilities

e.g. P(E) = P(E and M) + P(E and F)

= P(E|M)P(M) + P(E|F)P(F)
= 0.91 x 0.57 + 0.95 x 0.43 = 0.93

Independence Revisited

The rule for the intersection of two events is
P(X and Y) = P(X)P(Y|X) = P(Y)P(X|Y)

If P(X|Y) = P(X), then we would say X is independent of Y since the probability of X occuring is not affected by whether Y occurs or not. Substituting this into the equation above gives P(X and Y) = P(X).P(Y), the multiplication rule for independent events.

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