Question

A pack of cards was found to 51 cards if first 13 cards which are examined all are red then find the probability that the missing card is black

Answer 1

Hi there!
Here's the answer:

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SOLUTION:

^^ Using Bayes theorem, This problem can be solved

Let E1 the event that the missing card is black &
E2 be the event that the missing card is red.

Let A be the event the first 13 cards which are examined are all red.

•°• P(E1) = 1/2 & P(E2) = 1/2


P(A/E1 ) = Probability of selecting 13 red cards, when the missing card is black
= 26C13 / 51C13


P(A/E2 ) = Probability of selecting 13 red cards, when the missing card is red
= 25C13 / 51C13


The required probability is given by Bayes’ rule, by:

P(A|E)={P(A)×P(E/A)} ÷ {P(A)×P(E/A)+P(B)×P(E/B)}

= {(1/2)(26C13/51C13)} ÷ {(1/2)(26C13/51C13) + (1/2)(25C13/51C13)}

= {26C13} ÷ {26C13+25C13}

= {26!/(13!×13!)} ÷ {[26!/(13!×13!)] + [25!/(13!12!)]}

= {(26×25!)/(13×12!)} ÷ [{(26×25!)/(13×13!)] + [25!/(12!×13!)]

= {2} ÷ {(2+1)}

= 2/3

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:)


Hope it helps