# Probability of events not in use from pmf

#### Emekdavid

##### New Member
I have this question in a textbook I am reading and I am frustrated in how to answer it.
A mail order telephone business has 6 telephone lines. Let X = the number of lines in use at a specified time.
suppose the probability mass function, pmf, of X is given in the following table:
x | p(x)
0 | 0.1
1 | 0.15
2 | 0.20
3 | 0.25
4 | 0.20
5 | 0.06
6 | 0.04
The questions are:
a. Calculate the probability that between 2 and 4 lines inclusive, are not in use.
Here is what I did.
P(between 2 and 4 lines inclusive are not in use) = 1 - P(between 2 and 4 lines are in use) = 1 - P(2<= X <= 4)
P(2<=X<=4)=P(X=2, 3, or 4)= p(2) + p(3) + p(4) = 0.2 + 0.25 + 0.2 = 0.65
P(between 2 and 4 lines inclusive not in use) = 1 - 0.65 = 0.35
But the textbook says that the answer is 0.65. What is the problem or something that I am not missing out. That 0.65 is the probability that between 2 and 4 lines are in use. can someone help me where I am missing out on the calculation or my thinking?
Now the second one.

b. Calculate the probability that at least 4 lines are not in use.
I reasoned from this that if 4 lines are not in use, then that means at most 3 lines are in use. Then I decided to calculate the probability that at most 3 lines are in use. I got 0.7. But the textbook says I am wrong. It says the answer is 0.45. can someone help me with my reasoning and show me where I am wrong; how I could have done it better?
Thanks

#### katxt

##### Active Member
Your answers sound right to me. I had much the same problem with many textbooks when I was a student, and it's a real pain. kat