Question

In a hand at Whist, what is the probability that four kings are held by a specified player?

Answer 1

In a hand at whist. the specified player has 13 cards including 4 kings. Four kings can be drawn out of 4 in ${}^4{C}_4$ ways. The remaining 9 cards held by him can be drawn out of the remaining 48 cards in the pack in ${}^{48}{C}_9$ ways. The total number of ways in which 4 kings and 9 other cards can be drawn is ${}^4{C}_4{\times }^{48}{C}_9$

Number of favourable cases ${=}^4{C}_4{\times }^{48}{C}_9$
Number of exhaustive cases = ${}^{52}{C}_{13}$

$\therefore$ Required probability

$=\frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$

$=\frac{{}^4{C}_4{\times }^{48}{C}_9}{{}^{52}{C}_{13}}=\frac{11}{4165}$