Given:
A standard deck of 52 cards, has had all its face cards removed so that only the ace through ten of
each suit remains. A game is played in which two cards are drawn without replacement from this deck and a die is rolled. For the purpose of this game, an ace is considered to have a value of one.
Required:
To find the probability that the sum of the cards and the die is seven.
Explanation:
he sum of the cards and die is 7.
We have the same number of ways when the 2 cards are different (there is one roll that then adds to 7) and the same number of ways when the 2 cards are the same (once again, there is one roll that adds to 7.
If 2 cards are different, there are
[tex]C(4,1)\times C(4,1)=16ways[/tex]If 2 cards are the same, there are
[tex]C(4,2)=6[/tex]Note that the sum of the two cards must be less than 7, as the die is at least 1.
Different:(1,2), (1,3), (1,4),(1,5),(2,3),(2,4) Rolls are then 4, 3, 2, 1, 2, 1
The same: (1,1), (2,2), (3,3) - the roll is then 5, 3, and 1
[tex]\begin{gathered} =6\times16+3\times6 \\ =114 \end{gathered}[/tex][tex]\begin{gathered} =\frac{114}{4680} \\ \\ =\frac{19}{780} \end{gathered}[/tex]Final Answer:
19/780
