In a single tosses, we have
probability of getting a six =
1
6
, and probability of getting a non-six =(1-
1
6
)=
5
6
.
Let X denote the number of sixes in two tosses.
Then, clearly X can assume the value 0, 1, or 2.
P(X=0)=P(non-six in the 1st draw) and (non-six in the 2nd draw)]
=P(non-six in the 1st draw) ×P(non-six in the 2nd draw)
=(
5
6
×
5
6
)=
25
36
.
P(X=1)=P[six in the 1st draw and non-six in the 2nd draw) or (non-six in the 1st draw and six in the 2nd draw)]
=P(six in the 1st draw and non-six in the 2nd draw)
+P (non-six in the 1st draw and six in the 2nd draw)
=(
1
6
×
5
6
)+(
5
6
×
1
6
)=(
5
36
+
5
36
)=
10
36
=
5
18
.
P(X=2)=P[six in the 1st draw and six in the 2nd draw]
P=(six in the 1st draw) × P (six in the 2nd draw)
=(
1
6
×
1
6
)=
1
36
.
Hence, the probability distribution is given by
∴ mean,μ=Σxipi=(0×
25
36
)+(1×
5
18
)+(2×
1
36
)=
6
18
=
1
3
.
Variance, σ2=Σx
2
i
pi-μ2
=[(0×
25
36
)+(1×
5
18
)+(4×
1
36
)-
1
9
]=
5
18
.
Standard deviation, σ=
√
5
18
=
1
3
⋅
√
5
2
.
