Question

Four cards are chosen from a pack of cards without replacement. What is the probability that all four cards are hearts? What is the probability that all four cards are from red suits? What is the probability that all four cards are from different suits?

Answer 1

The total cards in a deck = 52, out of which 26 are black and 26 are red.

Among these 52 cards, Heart = 13 , Spade = 13 , Club = 13 and Diamond = 14

We will use the combination formula as:

$^nC_r = \frac{n!}{r!(n-r)!}$

1. Probability that all four cards are hearts

= $\frac{^{13}C_4}{^{52}C_4}$

= $\frac{13!}{9! 4!} \times \frac{48! 4!}{52!}$

= $\frac{17160}{6497400}$

= 0.0026

2. Probability that all four cards are from red suits

=$\frac{^{26}C_4}{^{52}C_4}$

= $\frac{26!}{22! 4!} \frac{4! 48!}{52!}$

= 0.055

3. Probability that all four cards are from different suits

= $\frac{^{13}C_1 \times ^{13}C_1 \times ^{13}C_1 \times ^{13}C_1}{^{52}C_4}$

= $\frac{13 \times 13 \times 13 \times 13}{270725}$

= 0.105