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