Suppose the event A has occurred. Provided the event A has occurred let us find the conditional probability of the event , i.e. .
By the probability multiplication rule
whence
Replacing by the total probability formula we obtain
.
Similarly we can obtain the formulas for the conditional probability of the other hypotheses.
Thus, provided the event A has occurred, the conditional probability of the event can be determined by the formula
which is known as Bayes’ formula.
The probabilities calculated by Bayes’ formula are often called probabilities of hypotheses.
Example 1. We have four boxes. The first box contains 2 red and 8 yellow balls, the second, 3 red and 2 yellow balls, the third,1 red and 1 yellow ball, and the fourth, 4 red and 6 yellow balls. The event is the selection of the i-th box (i=1,2,3,4). The probability of selecting of the i-th box is , i.e.
.
We randomly select one of the boxes and draw a ball from it. Calculate the probability of the ball being red.
Solution. Suppose the event A is the drawing of a red ball. If follows from the hypothesis that the conditional probability of drawing a red ball from the first box is equal to , i.e.
;
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