By Bayes'
theorem,
P(E3|A) = 
3
3
1
1
2
2
3
3
4
4
P(E ) P(A|E )
P(E )P(A|E )
P(E )P(A|E )+P(E )P(A|E )
P(E )P(A|E )
 
 
 
= 
1
1
4
7
0 165
1
3
1
1
1
1
1
4
4
18
4
4
4
7
4
13
×
=
×
+
×
+
×
+
×
Example 34 Find the mean of the Binomial distribution 
B 4, 31 Solution Let X be the random variable whose probability distribution is 
B 4,31 Here
n = 4, p = 1
3  and  q = 
1
2
1
3
3
−
=
We know that
P(X = x) =
4
4
2
1
C
3
3
x
x
x
−
⎛
⎞
⎛
⎞
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
, x = 0, 1, 2, 3, 4