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A card from a pack of 52 cards is lost. From the remaining car

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A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.

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A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.

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Text solutionVerified

Solution: let the event of lost card is diamond be A
let the event of both drawn card is diamond be B

P (A) = \frac{13}{52} = \frac{1}{4}

P^{'} (A) = 1 - \frac{1}{4} = \frac{3}{4}

P (\frac{B}{A}) = \frac{12}{51} \cdot \frac{11}{50}

P (\frac{B}{A ^{'}}) = \frac{13}{51} \cdot \frac{12}{50}

Now,

P (\frac{A}{B}) = \frac{P ( \frac{B}{A} ) . P ( A )}{P ( \frac{B}{A} ) . P ( A ) + P ( \frac{B}{A} ^{'} ) P ( A ^{'} )}

= \frac{\frac{12 \cdot 11}{51 \cdot 50} \cdot \frac{1}{4}}{\frac{12 \cdot 11}{51 \cdot 50} \cdot \frac{1}{4} + \frac{13 \cdot 12}{51 \cdot 50} \cdot \frac{3}{4}}

= \frac{12 \cdot 11 \cdot 1}{12 \cdot 11 \cdot 1 + 13 \cdot 12 \cdot 3}

= \frac{11}{11 + 39} = \frac{11}{50}

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Question Text	A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.
Answer Type	Text solution:1
Upvotes	150

A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.

Text solutionVerified

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