A card is drawn from a pack of 52 playing cards. The card is replaced and the pack is reshuffled. If this is done six times, the probability that 2 hearts, 2 diamonds and 2 black cards are drawn is
Answer & explanation
Correct answer: option 3
We have,
Probability of getting a heart in a draw $=\frac{13}{52}=\frac{1}{4}$
Probability of getting a diamond in a draw $=\frac{13}{52}=\frac{1}{4}$
Probability of getting a black card in a draw $=\frac{26}{52}=\frac{1}{2}$
In 6 draws, 2 draws must contains hearts, 2 draws must contain diamond cards and 2 draws must contain blcak cards. This can happen in $ {^6C}_2×{^4C}_2×{^2C}_2 $ mutually exclusive ways and the probability of each such way is $\left(\frac{1}{4}\right)^2 × \left(\frac{1}{4}\right)^2 ×\left(\frac{1}{2}\right)^2$
Hence, required probability
$= {^6C}_2×{^4C}_2×{^2C}_2 × \left(\frac{1}{4}\right)^2 × \left(\frac{1}{4}\right)^2 ×\left(\frac{1}{2}\right)^2= 90 ×\left(\frac{1}{2}\right)^{10}$