00:01
Okay, so we've got a company which charges £180 for insurance premium, and in the event of a fire, then they'll pay you £170 ,000.
00:10
And it tells the probability of the fire is 0 .1.
00:12
So the first thing it has to do is make a table for the payout, possible payouts in each policy.
00:17
So the payout is not including the $180 ,000 premium.
00:20
This is just what the company will pay out.
00:21
So they will pay out nothing if there's no fire, which is going to be obviously 0 .9 probability, and they'll pay out 170 ,000.
00:31
There is a fire, which there's a 0 .1 probability of.
00:34
Part b then asks us for the expected value, variance and sudden deviation of the profit.
00:39
The profit is now going to be 180 minus the payout.
00:45
So if there's no fire, probability of 0 .9, they're going to profit 180.
00:49
If there is a fire, they're going to lose 169 ,820 pounds.
00:59
I'm assuming, well, it might be in dollars, but yeah, whichever currency you're in.
01:03
And that has a probability of 0 .1 .1.
01:06
So the expectation of the profit is given by the sum of a value of the profit times the probability of having that profit.
01:19
So this is overall p.
01:21
So for instance, here, one profit, one possible profit is 180.
01:26
And the probability of that profit is 0 .9.
01:30
And another possible profit is minus 169 ,820.
01:36
And the probability of that is 0 .1.
01:39
And those are the only two values that the profit can take.
01:41
So you just do this and this, you find that this is minus 16 ,000, sorry, 820 pounds.
01:58
And the variance of the profit p is the expectation of the profit squared minus the expectation of the profit all squared like this.
02:13
And so the expectation of the profit squared is just the sum over p squared terms of the probability that p equals p.
02:25
And then we wrote down what the expectation of the profit was before, and so you just square that.
02:34
So if you do that, you'll find that this is roughly 2 .6 times 10 to the 9 variance, and the standard deviation is just the square root of the variance.
02:54
And that's 51 ,000.
02:59
Okay, then part c says, now suppose you issue two policies and the risk of fire is independent across the two policies and then it was to again make a table of the payouts.
03:15
So i'll say number of fires.
03:20
So the three possible payouts depending on how many fires there are.
03:28
There could be zero fire, so neither policy owner has a fire, one of them has a fire, or both of them have a fire.
03:36
The probability of neither of them having a fire is 0 .9 times 0 .9, which is 0 .81.
03:43
The probability of both of them having a fire is 0 .1 times 0 .1, which is 0 .0 .1.
03:49
And therefore, the probability of exactly one of them having a fire is 0 .18.
03:55
Oh, dear.
03:56
I've done that probability in the payoff...