Question

In a band at whist what is the chance that the 4 kings are held by a specific player?
Whist is a game of cards in which a standard pack of 52 cards are used. The game is played in Paris. Each round, a suit is randomly selected as ‘Trump’, which gets a preference over other suits for that particular round.

Answer 1

Hint:There are 52 cards in each pack. Find the number of cards for each player. One player should have 4 kings and 9 other cards. Find the favorable condition of the cards on that player. Then find the probability of finding 4 kings in the hand of 1 player.

Complete step-by-step answer:
We know that there are 52 cards in a pack. As there are 4 players, each player gets 13 cards.
i.e. $\dfrac{52}{4}=13$ cards for each player.
The number of possible distributions of 52 cards among 4 players are ${}^{52}{{C}_{13}}$.
Here combination is used because the order is not necessary.
The favorable condition is that one player should have all the 4 kings in his hand.
In a pack of cards there are 4 kings, 4 queens, 4 jacks and numbers from 2-10 and 4 aces.
One player should have 4 kings and 9 other cards to make the total number of cards in his hand 13.
Total number of cards excluding 4 kings from the pack of cards = 52 – 4 = 48 cards.
$\therefore$The favorable condition is ${}^{4}{{C}_{4}}\times {}^{48}{{C}_{9}}$.
$\therefore$The probability of finding 4 kings in the cards of one player
= ${}^{4}{{C}_{4}}\times {}^{48}{{C}_{9}}$ Distribution of card among 4 people = $\dfrac{{}^{4}{{C}_{4}}\times {}^{48}{{C}_{9}}}{{}^{52}{{C}_{13}}}$

Note: The favorable condition is that the specific player has 4 kings. Each player has 13 cards. So the 9 other cards should be taken from the rest 48 cards, excluding 4 kings.