If the coin is of gold, what is the probability that the other coin in the box is also of gold © NCERT
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 552
MATHEMATICS
Solution Let E1, E2 and E3 be the events that boxes I, II and III are chosen, respectively Then
P(E1) = P(E2) = P(E3) = 1
3
Also, let A be the event that ‘the coin drawn is of gold’
Then
P(A|E1) = P(a gold coin from bag I) = 2
2  = 1
P(A|E2) = P(a gold coin from bag II) = 0
P(A|E3) = P(a gold coin from bag III) = 1
2
Now, the probability that the other coin in the box is of gold
= the probability that gold coin is drawn from the box I = P(E1|A)
By Bayes' theorem, we know that
P(E1|A) =
1
1
1
1
2
2
3
3
P(E )P(A|E )
P(E )P(A|E )+P(E )P(A|E )+P(E )P(A|E )
=
1 1
2
3
1
1
1
1
3
1
0
3
3
3
2
×
=
× + × + ×
Example 18 Suppose that the reliability of a HIV test is specified as follows:
Of people having HIV, 90% of the test detect the disease but 10% go undetected