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 heart then the probability of the missing card to be a heart is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{11}{50}\)
Let \(E_1\) be the event that the lost card is a heart, \(E_2\) that it is not, and A that the two cards drawn from the remaining 51 are both hearts.
\(\displaystyle P(E_1) = \frac{13}{52} = \frac{1}{4},\quad P(E_2) = \frac{3}{4}\)
If a heart is lost, 12 hearts remain among 51 cards; otherwise 13 remain:
\(\displaystyle P(A|E_1) = \frac{^{12}C_{2}}{^{51}C_{2}} = \frac{66}{^{51}C_{2}},\quad P(A|E_2) = \frac{^{13}C_{2}}{^{51}C_{2}} = \frac{78}{^{51}C_{2}}\)
By Bayes’ theorem (the common factor \(^{51}C_{2}\) cancels):
\(\displaystyle P(E_1|A) = \frac{P(E_1)P(A|E_1)}{P(E_1)P(A|E_1) + P(E_2)P(A|E_2)}\)
\(\displaystyle = \frac{\frac{1}{4}\times 66}{\frac{1}{4}\times 66 + \frac{3}{4}\times 78}\)
\(\displaystyle = \frac{66}{66 + 234} = \frac{66}{300} = \frac{11}{50}\)
NCERT Exemplar:
Probability
