Problem 199SE
Messages that arrive at a service center for an information systems manufacturer have been classified on the basis of the number of keywords (used to help route messages) and the type of message, either email or voice. Also, 70% of the messages arrive via email and the rest are voice.
Determine the probability mass function of the number of keywords in a message.
Solution :
Step 1 of 1:
We assume that,
X is the number of keywords in the message.
E is the email message and
V is the voice mail message.
Then the table is given below.
Number of keywords 
0 
1 
2 
3 
4 
0.1 
0.1 
0.2 
0.4 
0.2 

Voice 
0.3 
0.4 
0.2 
0.1 
0 
Our goal is:
We need to find the probability mass function of the number of keywords in a message.
We know that 70% of the messages arrive via email and the rest are voice.
So P(V) = 0.3 and
P(E) = 0.7.
Range of x = [0, 1, 2, 3, 4].
And the probability of keywords in a mail either email or voice is given in the above table.
Using total probability theorem,
P(A) =
Then,
P(X=0) =
P(X=0) = 0.10.7+0.30.3
P(X=0) = 0.16
P(X=1) =
P(X=1) = 0.10.7+0.40.3
P(X=1) = 0.19
P(X=2) =
P(X=2) = 0.20.7+0.20.3
P(X=2) = 0.20
P(X=3) =
P(X=3) = 0.40.7+0.10.3
P(X=3) = 0.31
P(X=4) =
P(X=4) = 0.20.7+00.3
P(X=4) = 0.14
Then the probability mass function of the number of keywords in a message is
X 
0 
1 
2 
3 
4 
P(X=x) 
0.16 
0.19 
0.20 
0.31 
0.14 