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 there are 4 hearts is

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

Answer 1

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

The cases possible are:
(i) 3 H, 2 non-H: ${}^{13}{C}_3{\ast }^{39}{C}_2$
(ii) 4 H, 1 non-H: ${}^{13}{C}_4{\ast }^{39}{C}_1$
(iii) 5H: ${}^{13}{C}_5$
Hence, probability = $\frac{{}^{13}{C}_4{\ast }^{39}{C}_1}{{}^{13}{C}_3{\ast }^{39}{C}_2{+}^{13}{C}_4{\ast }^{39}{C}_1{+}^{13}{C}_5}$