Question

5 cards are drawn at random from a well shuffled pack of 52 playing cards. If it is known that there will be at least 3 hearts, the probability that all the 5 are hearts is

• A. $\frac{{}^{13}{C}_5}{{}^{52}{C}_5}$
• B. $\frac{{}^{13}{C}_5}{{}^{13}{C}_3{\times }^{39}{C}_2{+}^{13}{C}_4{\times }^{39}{C}_1{+}^{13}{C}_5}$
• C. $\frac{{}^{13}{C}_5}{{}^{13}{C}_3{+}^{13}{C}_4{+}^{13}{C}_5}$
• D. $\frac{{}^{13}{C}_5}{{}^{13}{C}_3{\times }^{13}{C}_4{\times }^{13}{C}_5}$

Answer 1

The correct option is B $\frac{{}^{13}{C}_5}{{}^{13}{C}_3{\times }^{39}{C}_2{+}^{13}{C}_4{\times }^{39}{C}_1{+}^{13}{C}_5}$

Let $P\left(k\right)$ = Probability of getting $k$ hearts among 5 drawn cards
$P\left(k\right)=\frac{{}^{13}{C}_k{\times }^{39}{C}_{5-k}}{{}^{5}2C_k}$
i.e choosing k cards out of 13 Heart cards and choosing remaining 5-k cards from others
$P(\ge 3.hearts)=P\left(3.hearts\right)+P\left(4.hearts\right)+P\left(5.hearts\right)$
$=\frac{{}^{13}{C}_3{\times }^{39}{C}_2{+}^{13}{C}_4{\times }^{39}{C}_1{+}^{13}{C}_5}{{}^{52}{C}_5}$
.
Now using definition of conditional probability
$P(k=5|k\ge 3)=P(k=5)/P(k\ge 3)$
$=\frac{{}^{13}{C}_5}{{}^{13}{C}_3{\times }^{39}{C}_2{+}^{13}{C}_4{\times }^{39}{C}_1{+}^{13}{C}_5}$