00:01
Okay, so here we're talking about rolling a fair four -sided die, and then whatever that dice says, it tells us how many times we flip a coin, and we record the total number of heads from the resulting coin flips, and we call that k.
00:16
So it says, suppose the die roll results in a two, and what's the conditional distribution for k, given that n equals two? so we want the possible values that k can take given the n equals two, and then the probabilities of those values.
00:32
So if we roll a two on the die, that means we flip a coin two times.
00:35
And we can either get zero heads, one head, or two heads.
00:40
Zero, each, it's a fair coin, so each flip, it's half probability of heads and half probability of tails.
00:46
So to get zero, we'd need to get tails twice, which has probability a quarter.
00:51
To get two, we need to get heads twice, which obviously has probability a quarter as well.
00:54
And then to get one, we could either get heads, tails or tails heads.
00:58
And so that's a quarter plus a quarter is a half.
01:02
So that's our answer for part a.
01:05
Part b then asks us to fill in the table of the joint pmf for n and k.
01:12
So along the top of the values of k and along down the side are the values of n.
01:31
So let's look at the case where an n equals zero.
01:36
The probability, because it's a fair die, so the probability that n equals 0, 1, 2 or 3 is just a quarter for each, okay, because they all got to be equal probabilities.
01:53
And so the probability that n is 0 and k is 0 is a quarter for n being 0.
01:59
And then if n is zero, we find that k has to be zero.
02:10
So you get a quarter times one or a quarter.
02:15
And then you can't get one, two or three heads if you flip no coins.
02:18
So those are all zero.
02:20
Then the probability for n to be one is a quarter.
02:22
The probability for k to be zero if n is one is just a half.
02:26
So that's going to give us an eighth here.
02:30
And similarly, if you roll a 1, the probability for that is a quarter and the probability of flipping one head is a half.
02:37
So that gives you an 8th as well.
02:39
And you can't flip two or three heads if you only flip a coin once.
02:44
Then to shake n equals 2 again, probability is a quarter.
02:48
If n equals 2, if you're throwing a coin two times, the probability to get 0, we saw up here was a quarter.
02:55
So we're going to have a quarter times a quarter, which would give us a 16th.
03:00
For one, it was a half, so we're going to have a quarter times a half, which is an eighth.
03:04
And again, two is going to get a 16th, and three is going to be zero.
03:07
Because if you only flip two coins, you can't get three heads.
03:10
Finally, for n equals three, the probability there is a quarter as well.
03:14
And then if you're flipping a coin three times the probability to get zero heads is an eighth.
03:20
So the joint probability of these two is a quarter times an eighth, which is one over 32.
03:25
Similarly, if you're flipping three coins, the probability to get one head is three over eight times the quarter from here.
03:34
That's three over 32.
03:36
That's the same as two heads.
03:39
And then three heads, you get one over 32.
03:42
So that's our joint pmf.
03:47
We're then asked to using this joint pmf...