Page 260 - Probability Demystified

250 APPENDIX

For two mutually exclusive events, A and B, where event B follows event A,

PðAÞ PðBjAÞ
PðAjBÞ¼
PðAÞ PðBjAÞþ PðAÞ PðBjAÞ

EXAMPLE: Box 1 contains two red balls and one blue ball. Box 2 contains
one red ball and three blue balls. A coin is tossed; if it is heads, Box 1 is
chosen, and a ball is selected at random. If the ball is red, find the probability
it came from Box 1.

SOLUTION:
Let A ¼ selecting Box 1 and A ¼ selecting Box 2. Since the selection of a box
1
is based on a coin toss, the probability of selecting Box 1 is and the proba-
2
1
1
1
bility of selecting Box 2 is ; hence, P(A) ¼ and P(A) ¼ . Let B ¼ selecting a
2 2 2
red ball and B ¼ selecting a blue ball. From Box 1, the probability of selecting
2
1
a red ball is , and the probability of selecting a blue ball is since there are
3 3
1
2
two red balls and one blue ball. Hence P(B|A) ¼ and PðBjAÞ¼ . Since there
3
3
1
is one red ball in Box 2, PðBjAÞ is , and since there are 3 blue balls in Box 2,
4
3
PðBjAÞ¼ . The probabilities are shown in Figure A-1.
4
Fig. A-1.

Hence
1 2 1

PðAÞ PðBjAÞ 2 3 3 8
PðAjBÞ¼ ¼ ¼ ¼
PðAÞ PðBjAÞþ PðAÞ PðBjAÞ 1 2 1 1 1 1 11
þ þ
2 3 2 4 3 8
In summary, if a red ball is selected, the probability that it came from
Box 1 is 8 .
11