00:01
So we're looking at a probability distribution.
00:04
I'm going to write out this distribution.
00:06
We have the values of x, which could be 1, 2, 3, 4, 5, 6, 7, 8 or 9.
00:14
And we're going to get the probabilities.
00:19
Now we've been given the formula.
00:22
Let's get those.
00:24
So we've got log of 2, log of 3 over 2, log of 4 over 3, and so on.
00:40
First of all, is it in fact a probability distribution? so first of all, we need to make sure that all the probabilities are between 1 and 0 inclusive.
00:52
So this is log base 10, and the largest one we're going to get here is log 2.
00:56
And yeah, log 2 is less than 1, because, oops, log base 10 of 2 is less than 1.
01:06
Of course it is because 10 to the power of x, of f of x, is 2.
01:15
It has to be something between 0 and 1.
01:22
So we do have that.
01:23
The other thing we need to check is if they sum to 1.
01:29
So we've got log 2, we've got log 3 over 2, 3 over 2, and so on, all the way up to the largest value here.
01:42
Or not actually the largest, but you get the idea.
01:46
Log 10 over 9.
01:50
Okay, so what is the sum of these probabilities? so can we add up some of these logs? i expect we can.
02:07
If we have all of them of the same base, we can use the product rule.
02:13
So i suppose we should do that.
02:15
The log product rule there is.
02:19
Since they are all of the same base, log 10 of ab is equal to log 10 of a plus log 10 of b.
02:32
If i apply here, log 10 of 2 times 3 over 2 times 4 over 3 all the way up 2, times 9 over 8 times 10 over 9.
02:47
Okay, you can see a lot of this is going to cancel out.
02:51
2 is going to cancel out, 3 is going to cancel out, and so on, all the way down to nine's counselling notes.
02:57
Leaving us with log base 10 of 10, which is 4.
03:01
So yes, it is a probability distribution.
03:04
These probabilities sum to 1.
03:07
Now let's get the actual values since we need to compare them.
03:14
Okay, so log 2 is 0 .301.
03:17
Does it say how many decimal places? it does not alters to 3 decimal places.
03:24
Log of 1 .5, 0 .176.
03:30
Log 4 over 3 .0 .0 .125.
03:37
Log 5 over 4.
03:41
0 .0 .097.
03:46
Log 6 over 5...