00:01
In this question, we have a switchboard that received an average number of calls in a 30 -minute period, and that is 13.
00:08
So we want to find the answers to a few parts here.
00:14
Now, before we do this, let's recognize that this is a poison distribution.
00:18
So we let x be the number of calls over a 30 -minute period.
00:31
So we know that x actually follows a poison distribution.
00:35
With the average or the mean of the thin.
00:40
So now for any poison distribution, where x is equals to r, we know that the formula is this, where the lambda is the mean.
00:58
So we can actually calculate our poison chances using this formula here.
01:08
Okay, now in part a, we want to find the probability that between 10 am to 10 .10.
01:15
There are 10 calls.
01:17
Now notice that this is lesser than 30 minutes.
01:20
In fact this is a 10 minute period.
01:22
So i'm going to let y be the number of calls over a 10 minute period.
01:37
So my y actually follows a poison distribution.
01:40
Now i'm going to take this 13 and divided it by 3 because 10 minutes is 1 third of 30 minutes so it's 13 over 3.
01:51
Now what we are wanting to find is the probability where y is equals to 10.
01:56
So i'm just subbing in the values here.
02:00
So my lambda is 13 over 3 to the power of r and that is 10 over r factorial.
02:07
So 10 factorial times e to power minus my lambda is 13 over 3.
02:13
So you just keen that into your calculator.
02:15
You get 0 .008443.
02:20
So this is getting exactly 10 calls.
02:25
Okay.
02:26
Now in part b, we want to find that.
02:29
Between 10 a .m.
02:31
To 10 .15 am...