00:01
It's given in this exercise that the average number of calls received by a switchboard in a 30 -minute period is 15.
00:11
So that's an average of 15 calls for 30 minutes.
00:16
And for a, b, and c we are asked for the probabilities of certain numbers of calls in a 30 -minute period.
00:23
When something is occurring randomly such as calls, and all we have is the rate of occurrence, we can model it with the posan random variable.
00:30
So if we say that the number of calls, if we call that x, then we can say x is a plus on random variable, and the mean is 15 for 30 minutes.
00:59
The probability mass function, or the plus on random variable, is given by this formula.
01:06
It's the mean to the exponent x times e to the minus mean over x factorial, where x is any non -negative integer.
01:21
And so for a we're dealing with 30 -minute period, what is the probability that we receive exactly 10 calls? so this is the probability that x is equal to 10.
01:38
So we have 15 to the exponent 10 times e to the minus 15 over 10 factorial.
01:52
And this comes out to approximately 0 .04 .86.
02:02
And then for b we are asked for the probability that we receive more than 9 calls but fewer than 15 calls in a 30 minute period.
02:12
So this is the probability that x is greater than 9 but less than 15.
02:26
Now we could solve this using the probability mass function manually by calculating the probability that x equals 10 plus the probability that x equals 11 plus the probability that x equals 12 and so on up to 14.
02:39
To be greater than 9, but less than 15, x can be 10, 11, 12, 13, or 14.
02:46
This can be done, but it's a bit tedious.
02:48
So let's use software.
02:50
So first let's re -express this as the probability that x is at most 14 minus the probability that x is at most 9.
03:04
These are equivalent statements...