Page 350 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Unit 27: Additive and Multiplicative Law of Probability

Note: (1) These two results hold only if P (A) > 0. Notes
(2) The results are proved under the tacit assumption that n is finite and that each sample
1
point has equal probability . It can be shown that the results hold in the general case
n
(without these restrictions).
Example 8: Two dice are thrown. Find the probability that the sum of the numbers in the two dice
is 10, given that the first die shows six.
Let A be the event that the sum of numbers in two dice is 10,
B be the event that the first die shows 6.
Then AB is the event that the sum is 10 and the first die shows 6 or which is equivalent
to the event that first die shows 6 and the second 4.
1 1
We have P (B) = , P (AB) =
6 36

P(AB) 1
Thus the required probability = P (A|B) = P(B) = . Thus the conditional probability
6
1
of getting a sum of 10, given that the first shows 6 is . The unconditional probability
6
1
of getting a sum of 10 is .
12
Example 9: Two coins are tossed. What is the conditional probability of getting two heads
(event B) given that at least one coin shows a head (event A) ?

3
Event A comprises of 3 sample points (HH), (HT), (TH) so that P (A) = ; the event
4
1
AB comprises of only one point (HH), so that P (AB) = . Thus the conditional
4
probability is

1
P(AB) 4 1
P (B|A) = = 3 = .
P(A) 3
4
Example 10: A box contains 5 black and 4 white balls. Two balls are drawn one by one without
replacement, i.e. the first ball drawn is not returned to the box. Given that the first ball
drawn is black, what is the probability that both the balls drawn will be black ?
1
Before the first draw the sample space consists of 9 points each with probability .
9
After the first draw the number of sample points reduces to 8 (as one ball is already
1
out of the box) and the probability of each sample point is .
8
5
Let A be the event that the first ball drawn is black then P (A) = , since there are 5
9
black balls.

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