00:01
Okay, so we're told that two sides show a single skull, which means something bad will happen.
00:09
So i'll say that the probability of rolling a skull is 2 out of 6, or a third.
00:17
And we also have two sides that show a single star, which equals one hit.
00:23
There's also one side that shows two hits, so the probability of rolling that is just 1 in 6.
00:29
And then there's one side which is blank, so the probability of rolling that is a 6 as well.
00:38
X is the number of hits rolled on a dice, and we want the expectation of x.
00:47
So x can either be...
00:51
There could be zero hits on the dice, which would be the case for the skull or the blanks.
00:57
And so the probability of that happening...
00:58
There's three sides which that could happen on, so the probability of that happening is a half.
01:02
You could get one hit, which we know the probability of is a third, because there are two different sides which you get one hit.
01:09
Or you could get two hits, and the probability of that happening is a sixth.
01:13
So that's our distribution for x.
01:15
And so the expectation is just the sum of x times the probability that it equals that value.
01:19
And so we get 0 times a half, plus 1 times a third, plus 2 times a sixth, which is equal to 2 thirds.
01:37
The variance of x is then given by the expectation of x squared minus the expectation of x squared.
01:48
So we need the expectation of x squared, which, just replacing these numbers by their squares, we're going to get a third plus...
01:59
1 times a third plus 4 times a sixth, which is just going to be 1.
02:06
And so the variance is equal to 1 minus 2 thirds squared, which is 5 over 9.
02:19
The next bit asks for the expectation and variance of three dice rolls.
02:30
Now each of the dice rolls are going to be independent, so the expectation of three dice rolls is just the expectation of 3x, which is 3 times the expectation of x by the linearity of expectation, which is just 2.
02:43
And the variance, when we bring out a factor in front of the variance, we get a square.
02:52
So it's going to be 5.
02:57
The next bit, i'll call that part 2, i'll call this part 3, the next bit asks for the expectations and variances if you change the blank to a hit, or if you change the double hit to a triple hit.
03:13
So if you change the blank side to a hit, i'll call the number of hits in this case xb to h, because you've changed the blank side to a hit, and we were looking at the number of hits.
03:25
Then the distribution for xb to h, again, you can get 0 hits, 1 hit or 2 hits.
03:32
But this time, the probability to get 0 hits is, instead of a half, it's a third, because your blank has gone.
03:44
The probability to get 1 hit is a half, and to get 2 is a sixth, because now you've got three sides where you can get 1 hit.
03:53
And on the alternative option where you can change your double hit to a triple hit, the number of hits you can get is now 0, 1 or 3, and the probabilities are a half, a third and a sixth...