Question

Seven cards are selected from a standard deck of cards. a) Determine the probability that exactly 5 of them are hearts. b) Determine the probability that there are 3 hearts and 3 diamonds. c) Determine the probability that there are 3 hearts, 3 diamonds and 1 spade. d) Determine the probability that there are 2 Aces and 2 Kings. e) Determine the probability that there are 2 Aces and 3 Kings.

Answer 1

a

`P(A_1)  =  0.007`

b

`P(A_2) = 0.016`

c

`P(A_3) =  0.008`

d

`P(A_4) =  0.004`

e

`P(A_5) =  0.0002`

Explanation:

From the question we are told that

The number of cards selected is n = 7

Generally in a standard deck of cards

The total number of cards is N = 52

The number of hearts is h = 13

The number of diamonds is d = 13

The number of spade is s = 13

The number of Ace is a = 4

The number of kings is k = 4

Considering question a

The number of ways to selected 5 hearts out of the 13 hearts is mathematically represented as

`G  =  ^(13)C_5`

Here C means combination so

The number of cards that is not hearts is 52 - 13 = 39

Generally the number of ways of selecting the reaming 2 cards is

`H  =  ^(39)C_2`

Generally the number of ways to select the 7 cards from the 52 deck of cards is

`V  =  ^(52)C_7`

Generally the probability that exactly 5 of the 7 cards are hearts is mathematically represented as

`P(A_1)  =  (G  *  H)/(V)`

=>
`P(A_1)  =  (^(13)C_5  *  ^(39)C_2)/(^(52)C_7)`

=>
`P(A_1)  =  0.007`

Considering question b

The number of ways to selected 3 diamonds out of the 13 diamonds is mathematically represented as

`B =  ^(13) C_3`

The number of ways to selected 3 hearts out of the 13 hearts is mathematically represented as

`K  =  ^(13)C_3`

The number of cards that is not hearts or diamond is 52 - (13 +13) = 26

Generally the number of ways of selecting the reaming 1 cards is

`M  =  ^(26)C_1`

Generally the probability that there are 3 hearts and 3 diamonds is

`P(A_2) =  (B  *  K *  M)/(V)`

=>
`P(A_2) =  (^(13)C_3  *  ^(13)C_3 *  ^(26)C_1)/(^(52)C_7)`

=>
`P(A_2) = 0.016`

Considering question c

The number of ways to selected 1 spade out of the 13 spade is mathematically represented as

`J = ^(13)C_1`

Generally the probability that there are 3 hearts, 3 diamonds and 1 spade is

`P(A_3) =  (B *  K  *  J)/(V)`

=>
`P(A_3) =  (^(13)C_3 *  ^(13)C_3   *  ^(13)C_1)/(^(52)C_7)`

=>
`P(A_3) =  0.008`

Considering question d

The number of ways to selected 2 Aces out of the 4 Aces is mathematically represented a

`U = ^(4)C_2`

The number of ways to selected 2 Kings out of the 4 Kings is mathematically represented a

`R = ^(4)C_2`

The number of cards that is not Aces or Kings is 52 - (4 +4) = 44

Generally the number of ways of selecting the reaming 3 cards is

`S = ^(44)C_3`

Generally the probability that there are 2 Aces and 2 Kings is

`P(A_4) =  (U *  R  *  S)/( V)`

=>
`P(A_4) =  ( ^(4)C_2 *  ^(4)C_2  *  ^(44)C_3)/( ^(52)C_7)`

=>
`P(A_4) =  0.004`

Considering question e

The number of ways to selected 3 Kings out of the 4 Kings is mathematically represented a

`P = ^(4)C_3`

Generally the number of ways of selecting the reaming 2 cards is

`Q = ^(44)C_2`

Generally the probability that there are 2 Aces and 3 Kings is

`P(A_5) =  (U  *  P  *  Q )/(V)`

=>
`P(A_5) =  ( ^(4)C_2  *  ^(4)C_3  *  ^(44)C_2 )/(^(52)C_7)`

=>
`P(A_5) =  0.0002`