DRV Probability

Question

Suppose we have a standard deck of 52 cards (not including jokers). We are playing 5-card poker where players are handed 5 cards each. Find the probability all 5 cards are black. And find the probability we have exactly 3 diamonds

Answer 1

for all 5 cards being black you can take

P(5 black cards) = C(26,5) / C(52,5) = 65,780 / 2,598,960 = 0.0253

meaning that the probability that all 5 cards are black is approximately 2.53%

Answer 2

52 standard deck of cards, there are 26 blacks & 26 reds, 13 hearts & 13 diamonds & 13 spades & 13 clubs,

A, 2, 3, 4, . . . , 10, J, Q, K

l Number cards l Face cards

Now, 5 cards are all black.

1. This is combination, which means that order does not matter

ex) I get A, 2, 3, 4, J or I get 2, 3, 4, A, J . Both consider the same hand.

1. There are 26 blacks.

26 C 5 = 65780 ways to get ( 26 blacks and we receive 5 blacks from 26 blacks)

1. P(all blacks) = 65780 / (52 C 5) = 65780/ 2598960 = .025310124 ~~ .0253

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Exactly 3 diamonds, means 3 diamonds + 2 non-diamonds

1. There are 13 diamonds and 39 non-diamonds (52 - 13 = 39)
2. exactly 3 D = 13 C 3 * 39 C 2 = 211926
3. P(exactly 3 D) = 211926 / 2598960 = .081542617 ~~.0825

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Detail:

there are 52 cards and we receive 5 cards => total number of receiving cards is 52 C 5.

exactly 3 diamonds = 13 C 3 * 39 C 2

=> 13 diamonds and we receive 3 of them /// 39 non diamonds and we receive 2 of them

So, 1. P(all blacks) = .0253 , 2. P(exactly 3 D) = .0825