Suppose that a deck of 52 cards contains 26 red cards and 26 black cards. Say...

Suppose that a deck of 52 cards contains 26 red cards and 26 black cards. Say we use the 52 cards to randomly distribute 13 cards each among two players (2 players receive 13 cards each). a. How many ways are there to pass out 13 cards to each of the two players? b. What is the probability that player 1 will receive 13 cards of one color and player 2 receive 13 cards of the other color?

Solutions

Expert Solution

52 cards contains 26 red cards and 26 black cards.

We randomly distribute 13 cards each among two players (2 players receive 13 cards each).

(a) Ways to pass out 13 cards to each of the two players :-

The first player can receive 13 cards out of the possible 52 in 52C13 ways.

The second player can then receive 13 cards out of the possible (52-13)=39 in 39C13 ways.

Moreover the two players can also be arranged among themselves i.e. the set of cards that Player 1 has can be with Player 2 = 2! ways

Total ways to pass out 13 cards to each of the two players = 52C13 * 39C13 * 2!

(b) Probability that player 1 will receive 13 cards of one color and player 2 receive 13 cards of the other color.

The first player can receive 13 cards out of the possible 26 in 26C13 ways.

The second player can then receive 13 cards out of the possible 26 in 26C13 ways.

Moreover the two players can also be arranged among themselves i.e. the set of cards that Player 1 has can be with Player 2 = 2! ways

Total ways of selecting 26(13+13) cards out of 52 is 52C26

Thus required probability = 26C13 * 26C13 * 2! / 52C26 = 0.43625

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Thanks!

orchestra answered 1 year ago

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