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Bayes' Theorem says that:
Note that the union of all of the As (A1, A2, ... An) = the total sample space, so they cover every possibility.
There are two bags containing balls of various colours. A bag is selected at random and a ball taken from it at random. The probability of picking a blue ball out of bag 1 is ½ . The probability of picking a blue ball from bag 2 is ¼ . If the experiment is carried out and a blue ball is selected, what is the probability that bag 2 was selected?
Let A2 be the event that bag 2 was selected and let A1 be the event that bag one was selected. Let B be the event that a blue ball is chosen. Then, using Bayes' Theorem:
P(A2|B) = P(B|A2)P(A2) .
P(B|A1)P(A1) + P(B|A2)P(A2)
Now, P(B|A2) = probability of picking a blue ball given that bag A2 is selected = ¼ from the question.
Similarly, P(B|A1) = ½ .
P(A1) = probability of selecting bag 1 = ½ = P(A2)
Hence P(A2|B) = ¼ × ½.
½ × ½ + ¼ × ½