00:01
So we have a scenario where we have a coin is tossed and we're going to assume the coin is, you know, the probability of heads equals that of tails, which will be a half.
00:12
I'll just go ahead and make that assumption.
00:14
And then if it lands heads, then you're going to roll an ordinary six -sided die, one through six.
00:20
But if it lands tails, you're, we're going to roll a special die, and the special die has six faces, has only the faces two, two, three, three, three, and four.
00:35
So still six sides, however, the numbers are different.
00:40
And the random variable x, we're going to say x is the number rolled on the die.
00:47
So x will take values zero, one, two, whoops, not zero, sorry, sorry.
00:54
That's not zero on the six -sided dice.
00:56
One, two, three, four, five, and six.
01:01
And we want to find the probability distribution for x.
01:05
And the fact that we have this extra dice in here with the different faces will change the probability distribution for x.
01:14
Because normally, if we just look at a regular old die, one through six, it's an equal probability of occurring.
01:28
This is not the probability here.
01:30
This, my, the probability would actually be one -sixth for each outcome.
01:36
That's our probability distribution.
01:39
But we, we have to think about what happens if we roll heads.
01:43
Well, to roll heads, we have to get one -half.
01:45
So the probability that you get, let's make a little decision tree.
01:49
We go, we can go heads or tails, right? heads and tails.
01:54
If we go heads, then we have our regular old six -sided die with its regular six outcomes, right? this is a half.
02:02
Each of these six branches will have a probability of a sixth.
02:07
And so what we get is essentially a twelfth.
02:10
We multiply across.
02:11
So one -sixth multiplied by a half all the way down is going to give us the probability of heads.
02:21
So this is the probability of, of, of rolling a, rolling a one, given you flipped heads is 0 .083 repeating, or a twelfth.
02:30
And it's all the way down along here.
02:33
But we have this other probability with tails.
02:35
Well, let's think about tails.
02:37
Well, that's still one -half, but then these probabilities will be different here.
02:40
Because if we, for a two on this other die, so the probability of two given on this other die is one -third, two out of six, two out of six.
02:55
I don't like to, i tell my students not to round, so i won't round.
02:58
The probability of three is going to be three out of six, because there's three ways you can get that.
03:02
And then the probability of four, well, that's just one out of six.
03:06
And then the probability of one, five, or six are all zero, because you can't get one, five, or six on this second die.
03:17
So what we do for these values, for two, three, and four, well, what we do is we take the, to get the tails probabilities, we take each of these, two -sixths, three -sixths, and one -sixth, and multiply those by a half.
03:46
There we go.
04:00
So we're going to take each of these values, boom, boom, boom, and zero, i mean, we can ignore those, because that is zero, anything times zero is zero.
04:08
So what we do is we just take these values and multiply them by a half.
04:11
So two given heads, probability of two given, or excuse me, given tails, well, that's two -sixths times a half, which is one -sixth.
04:21
Probability of three given tails, that's going to be, or i should say two and tails, and tails, and tails, three and tails, that's going to be three -sixths times one -half.
04:42
And then probability of four and tails, that's one -sixth times a half.
04:52
And that's what we have here.
04:54
Two is, there's the one -sixth, this is going to be a quarter, right, because this is going to be three over twelve, which is a quarter, that's where the 0 .25 is for the probability of three, given the tails, or under the tails column here.
05:12
And then the one -twelfth, we've seen that before, the one -sixth times a half, there's the twelfth right there.
05:16
So this is, essentially we have a joint distribution table.
05:20
We have our values here for x, one through six, and our heads, right, so this is a joint distribution table...