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Statistics

Notes Here P(A | H) is read as the conditional probability of A given the event H. Note that we can
write

1/9 P(A  H)
P(A | H) = 
5/9 P(H)
This discussion enables us to introduce the following formal definition. In what follows we
assume that we are given a random experiment with discrete sample space R, and all relevant
events are subsets of R.

3.1 Conditional Probability

Definition 3 : Let H be an event of positive probability, that is, P(H) > 0. The conditional
probability P(A | H) of an event A, given the event H, is
P(A  H)
P(A | H) = ...(9)
P(H)
Notice that we have not put any restriction on the event A except that A and H be subsets of the
same sample space R and that P(H) > 0.

Now we give two examples to help clarify this concept.

Example 12: In a small town of 1000 people there are 400 females and 200 colour-blind
persons. It is known that ten per.cent, i.e. 40, of the 400 females are colour-blind. Let us find the
probability that a randomly chosen person is colour-blind, given that the selected person is a
female.
Now suppose we denote by A the event that the randomly chosen person is colour-blind and by
H the event that the randomly chosen person is a female. You can see that
P(A  H) = 40/1000 = 0.04 and that
P(H) = 400/1000 = 0.4.

Then
P(A  H) 0.04
P(A | H) =   0.1.
P(H) 0.40
Now can you find the probability that a randomly chosen person is colour-blind, given that the
selgcted person is a male?
If you denote by M the event that the selected person is a male, then

600
P(M) =  0.6 and
1000

600
P(AM) =  0.16.
1000
0.16
Therefore, P(A | M) =  0.266.
0.6
You must have noticed that P(A | M) > P(A | H). So there are greater chances of a man being
colour-blind as compared to a woman.

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