Problem 2
Four cards are selected at random without replacement from a well-shu
ffl
ed deck of 52 playing cards. Find the probability
that
(A)
all of the cards are hearts?
Solution:
There are 13 hearts and we want 4 of them.
n
(
E
) =
C
(13
,
4)
n
(
S
) =
C
(52
,
4)
P
(
E
) =
C
(13
,
4)
C
(52
,
4)
≈
.
0026
(B)
exactly 2 of the cards are diamonds?
Solution:
There are 13 diamonds and we want 2 of them. There are 39 non-diamonds and we want 2 of them (since
we need a total of 4 cards).
n
(
E
) =
C
(13
,
2)
C
(39
,
2)
n
(
S
) =
C
(52
,
4)
P
(
E
) =
C
(13
,
2)
C
(39
,
2)
C
(52
,
4)
≈
.
2135
(C)
two of the cards are face cards (i.e. king, queen, jack) and the other two cars are fours?
Solution:
There are 12 face cards and we want 2 of them. There are 4 fours and we want two of them.
n
(
E
) =
C
(12
,
2)
C
(4
,
2)
n
(
S
) =
C
(52
,
4)
P
(
E
) =
C
(12
,
2)
C
(4
,
2)
C
(52
,
4)
≈
.
0015
(D)
none of the cards are red cards?
Solution:
There are 26 red cards and we want 0 of them. There are 26 black cards and we want 4 of them.
n
(
E
) =
C
(26
,
0)
C
(26
,
4)
n
(
S
) =
C
(54
,
4)
P
(
E
) =
C
(26
,
0)
C
(26
,
4)
C
(52
,
4)
≈
.
0552
(E)
one of the cards is a club, two of the cards are spades, and the other card is a diamond?
Solution:
There are 13 clubs and we want 1 of them.
There are 13 spades and we want 2 of them.
There are 13
diamonds and we want 1 of them.
n
(
E
) =
C
(13
,
1)
C
(13
,
2)
C
(13
,
1)
n
(
S
) =
C
(52
,
4)
P
(
E
) =
C
(13
,
1)
C
(13
,
2)
C
(13
,
1)
C
(52
,
4)
≈
.
0487
2