00:01
We're looking at a lot of 100 units, rather a sample of 100 units, and they will be rejected if three or more units are defective.
00:10
So for part a, we are looking at our 100 units, n is 100.
00:16
2 % are expected to be defective.
00:19
So the probability of each of these being defective is 0 .02.
00:24
What is the probability that 3 or more, so x the number of defective is at least 3? what's that probability of that many being defective? okay, so what we have here is a binomial distribution.
00:42
We have 100 independent trials.
00:45
We have two outcomes.
00:46
Each unit is defective or not.
00:48
And we have the same probability for each, because we have this rate.
00:53
So what i'm going to need to use is the binomial formula.
00:57
Probability of x being defective, n choose x, p to the x, 1 minus p to the n minus x.
01:05
This gives me the probability of exactly x being defective.
01:09
So to get at least 3, i could find the probability that 3 is effective, then 4, then 5, all the way up to 100.
01:16
That would take a really, really long time.
01:19
So i'm not going to do that.
01:20
I'm going to use the complement rule.
01:24
So the complement rule, which i will note down, is looking at a really, really long time.
01:29
Be 100 units and recognising the x can only take 101 different values.
01:35
It could be 0, defective, it could be 1, 2, all the way up to 100.
01:40
One of those is definitely going to happen.
01:42
We have a probability distribution here, so if you add up all those probabilities, you get a total of 1.
01:49
The complement rule is where you start at 1 and you subtract the outcomes you don't want.
01:56
If i'm looking at 3 or more, so i don't want, is two or fewer.
02:04
So i'm going to find this which will be a lot faster because this can only be 0, 1 or 2.
02:12
I'm going to start with 2 to demonstrate this formula.
02:16
So we have 100 independent trials.
02:19
So if i want to multiply the probability for different trials i can just multiply them.
02:23
This term is for the 2 that are defective.
02:26
A probability of 0 .02.
02:29
It happens twice.
02:31
So if it's multiplied by itself.
02:33
This term is the ones that are not defective.
02:37
Probability of 0 .98, 98 % of them aren't defective, and it happens 98 times.
02:46
But here i have the probability that the first unit i can't get at is defective...