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Standard XII
Mathematics
Fundamental Principle of Counting
Three cards a...
Question
Three cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of spades. Hence, find the mean of the distribtution. [CBSE 2015]
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Solution
We
have
,
p
=
probability
of
getting
a
spade
in
a
draw
=
13
52
=
1
4
and
q
=
1
-
p
=
1
-
1
4
=
3
4
Let
X
denote
a
success
of
getting
a
spade
in
a
throw
.
Then
,
X
follows
binomial
distribution
with
parameters
n
=
3
and
p
=
1
4
∴
P
X
=
r
=
C
r
3
p
r
q
3
-
r
=
C
r
3
1
4
r
3
4
3
-
r
=
C
r
3
3
3
-
r
4
3
=
27
64
C
r
3
3
r
,
where
,
r
=
0
,
1
,
2
,
3
So
,
the
probability
distibution
of
X
is
given
by
:
P
X
=
r
=
27
64
C
r
3
3
r
,
where
,
r
=
0
,
1
,
2
,
3
Now
,
Mean
,
E
X
=
n
p
=
3
×
1
4
=
3
4
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