Question

52 well-shuffled playing cards are distributed at random among 4 player's.Find the probability that a hand contain (i) 3 aces (ii) 2 kings and one queen (iii) No cards of diamond (iv) 5 pictured cards.

Answer 1

(i) Out of 13 cards 3 can be selected in $C^{13}_3$ ways = 286

Out of 4 aces 3 can be selected in $C^4_3$ ways = 4

Probability of selecting 3 aces:

$= \frac{C^4_3}{C^{13}_3}=\frac{4}{286}=0.0139$

(ii) Out of 4 kings 2 can be selected and out of 4 queens 1 can be selected as:

$C^4_2 \times C^4_1 = 6 \times 4 = 24$

Probability of getting 2 kings and 1 queen:

$= \frac{C^4_2 \times C^4_1}{C^{13}_3} = \frac{24}{286}=0.0839$

(iii) Out of 52, diamond cards are 13, So, non-diamond cards are 52-13 = 39

Probability of selecting no cards of diamond:

$\frac{C^{39}_{13}}{C^{52}_{13}}=0.0128$

(iv) Out of 12 pictured cards 5 can be selected as $C^{12}_5$ ways.

Probability of getting 5 pictured cards:

$= \frac{C^{12}_5}{C^{13}_5}=0.6154$