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:

a

[tex]P(A_1)  =  0.007[/tex]

b

[tex]P(A_2) = 0.016[/tex]

c

[tex]P(A_3) =  0.008[/tex]

d

[tex]P(A_4) =  0.004[/tex]

e

[tex]P(A_5) =  0.0002[/tex]

Step-by-step 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

      [tex]G  =  ^{13}C_5[/tex]

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  

    [tex]H  =  ^{39}C_2[/tex]

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

        [tex]V  =  ^{52}C_7[/tex]

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

     [tex]P(A_1)  =  \frac{G  *  H}{V}[/tex]

=>  [tex]P(A_1)  =  \frac{^{13}C_5  *  ^{39}C_2}{^{52}C_7}[/tex]

=>   [tex]P(A_1)  =  0.007[/tex]

Considering question b

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

      [tex] B =  ^{13} C_3 [/tex]

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

      [tex]K  =  ^{13}C_3[/tex]

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  

    [tex]M  =  ^{26}C_1[/tex]

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

    [tex]P(A_2) =  \frac{B  *  K *  M}{V}[/tex]

=> [tex]P(A_2) =  \frac{^{13}C_3  *  ^{13}C_3 *  ^{26}C_1}{^{52}C_7}[/tex]

=>  [tex]P(A_2) = 0.016[/tex]

Considering question c

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

     [tex]J = ^{13}C_1[/tex]

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

   [tex]P(A_3) =  \frac{B *  K  *  J}{V}[/tex]

=> [tex]P(A_3) =  \frac{^{13}C_3 *  ^{13}C_3   *  ^{13}C_1}{^{52}C_7}[/tex]

=>  [tex]P(A_3) =  0.008[/tex]

Considering question d

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

     [tex]U = ^{4}C_2[/tex]

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

       [tex]R = ^{4}C_2[/tex]

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  

     [tex]S = ^{44}C_3[/tex]

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

    [tex]P(A_4) =  \frac{U *  R  *  S}{ V}[/tex]

=>  [tex]P(A_4) =  \frac{ ^{4}C_2 *  ^{4}C_2  *  ^{44}C_3}{ ^{52}C_7}[/tex]

=>[tex]P(A_4) =  0.004[/tex]

Considering question e

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

     [tex]P = ^{4}C_3[/tex]

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

      [tex]Q = ^{44}C_2[/tex]

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

     [tex]P(A_5) =  \frac{U  *  P  *  Q }{V}[/tex]

=>  [tex]P(A_5) =  \frac{ ^{4}C_2  *  ^{4}C_3  *  ^{44}C_2 }{^{52}C_7}[/tex]

=>  [tex]P(A_5) =  0.0002[/tex]

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