Correct Answer - Mean =
3
4
,variance =
9
16
There are 13 hearts and 39 other cards.
Let E= event of drawing a heart.
Then, P(E)=
13
52
=
1
4
andP(
ˉ
E
)=(1-
1
4
)=
3
4
.
Let X= number of hearts in a draw.
Then, X=0,1,2 or 3.
P(X=0)=P(
ˉ
E
ˉ
E
ˉ
E
)=P(
ˉ
E
)×P(
ˉ
E
)×P(
ˉ
E
)=(
3
4
×
3
4
×
3
4
)=
27
64
.
P(X=1)=P[(E
ˉ
E
ˉ
E
)or(
ˉ
E
ˉ
E
E)]
=P(E
ˉ
E
ˉ
E
)+P(
ˉ
E
E
ˉ
E
)+P(
ˉ
E
ˉ
E
E)
=(
1
4
×
3
4
×
3
4
)+(
3
4
×
1
4
×
3
4
)+(
3
4
×
3
4
×
1
4
)=
27
64
.
P(X=2)=P[(EE
ˉ
E
)or(E
ˉ
E
E)or(
ˉ
E
EE)]
=P(EE
ˉ
E
)+P(E
ˉ
E
E)+P(
ˉ
E
EE)
=(
1
4
×
1
4
×
3
4
)+(
1
4
×
3
4
×
1
4
)+(
3
4
×
1
4
×
1
4
)=
9
64
.
P(X=3)=P(EEE)=P(E)×P(E)×P(E)=(
1
4
×
1
4
×
1
4
)=
1
64
.
Find the mean and variance.