Question

A flush in the card game of poker occurs if a player gets five cards that are all the same suit​ (clubs, diamonds,​ hearts, or​ spades). Complete parts​ (a) and​ (b) to obtain the probability of being dealt a flush in five cards. ​(a) Initially concentrate on one​ suit, say spades. There are 13 spades in a deck. Compute​ P(five spades​)equals​P(first card is spades and second card is spades and third card is spades and fourth card is spades and fifth card is spades​). ​P(five spades​)equals nothing

Answer 1

The probability is 0.000495

Explanation:

As per the question:

Total no. of cards in a deck = 52

No. of spades in a deck = 13

Now, we have to select 5 cards in a deck such that they belong to the same suit, i.e., spades.

The no. of ways of selecting 5 cards from a deck =
`^(52)C_(5) = (52!)/(5!(52 - 5)!) = (52!)/(5!47!)`

The no. of ways of selecting 5 cards from 13 spade cards =
`^(13)C_(5) = (13!)/(5!(13 - 5)!) = (13!)/(5!8!)`

Now,

Probability that the selected 5 cards are all spades, P(E) =
`(No.\ of\ ways\ of\ selecting\ 5\ cards\ from\ 13\ spade\ cards)/(No.\ of\ ways\ of\ selecting\ 5\ cards\ from\ a\ deck)`

P(E) =
`(^(13)C_(5))/(^(52)C_(5))`

P(E) =
`((13!)/(5!8!))/((52!)/(5!52!))`

P(E) =
`((13* 12* 11\tiems 10* 9)/(5* 4* 3* 2* 1))/((52* 51* 50* 49* 48)/(5* 4* 3* 2* 1))`

P(E) =
`(13* 12* 11\tiems 10* 9)/(52* 51* 50* 49* 48) = 4.95* 10^(- 4) = 0.000495`