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0:04
hi in this video we're going to do a couple of examples uh one's going to be a law total probability example another is going to be a bayes theorem example but as we've seen before they're very closely related okay so let's just jump right in uh both examples use the same sort of uh format here where we have two urns earn one contains 15 red balls and 10 green balls and earn two contains uh four red balls and and 15 green balls a ball is going to be selected uh
0:32
randomly selected from earn one and transferred to earn two and then a ball is going to be selected from earn two after after a ball has been transferred we're going to select a ball from urn two and we want to know what the probability is that the ball selected from urn two is red okay so before we get into kind of you know the mathematics and the c and the notation and all behind this uh let's just kind of logically think through it and hopefully develop some
0:56
intuition or maybe you already have the intuition so we're gonna transfer a ball from uh from earn one to earn two and of course that ball could be red or or green uh the ball would be red with probability since since there are 15 red balls out of 25 total balls and earn one the probability that the balls transferred from earn one to earn two as read as 15 out of 25 or 0.6 and likewise the probability that the ball was green would be 10 out of 25 or 0.4 and so just
1:27
logically with probability 0.6 then earn 2 is going to have 5 red balls and 15 green balls because with that probability 0.6 a ball a red ball was added to earn two and so with that probability of 0.6 the probability then of selecting a red ball from earn 2 would be 5 out of 20. there are 5 red balls out of the 20 total and that probably that that fraction is is 1 4 or zero point two five in a similar way with probability point four uh the urn two then would look
2:01
would have four red balls and sixteen green balls we've transferred a green ball over and so with probability point four then the probability of of of selecting a a red ball from urn two would be four out of twenty there would be four red balls now out of twenty or zero point two and so then it's just uh you know uh a weighted average then i take a weighted average of the 0.25 and the point and the and the 0.2 of course which is what a a uh a law of total probability is so the
2:30
probability that a red ball is selected from earned two after i transferred a ball from earn one would be the 0.25 times 0.6 and then plus 0.2 times 0.4 so we wait that we weight the values 0.25 and 0.2 with the weights of 0.6 and 0.4 and we get a 0.23 okay so now let's add a little uh mathematic mathematical notation to this so let's start by defining the event cap t to be the event that the color of the uh i'm sorry divine capped t to be the color of the uh of the transferred ball and so
3:06
we have the probability that cap t is red is 0.6 and the probability that cap t is green is 0.4 okay so now let's add another letter here to a cap c to represent the uh the color of the ball uh of the ball selected from urn two and so what we have then is mathematically how we would write this in is that we would have seen these conditional probabilities the probability that the color of the ball from earned two is red given that the transfer ball was red is
3:40
five out of twenty and the probability that uh the color of the ball selected from earn2 is red given that the transfer ball was green was 4 out of 20. this is all what we did before we're just kind of putting some some mathematical symbols to it and of course the 5 out of 20 and the 4 out of 20 are these decimal values and then as i said before this is nothing more than a law of total probability problem i would i would calculate the probability that the ball selected
4:09
from earned 2 is read by taking 0.25 times that 0.6 plus the point 2 times the times the 0.4 okay so that's a law total probability problem i need to uh in order to get to the next problem i need to uh develop this notation and and this is the notation that we have okay so now let's look at the next problem uh the first two lines in example two are exactly the same as the first two lines in example one we have the exact same set up but now look at the third line
4:37
this time we're going to say the ball selected from earn two is red we're going to be given that the ball selected from earn 2 is red and we want to calculate now the probability that the transferred ball was green that's a little bit harder problem so let me go back to example one so i'm going to show you example one and all i'm doing all i want to point out is this last line in the in the wording of the problem this was example one we were asked to calculate the probability uh
5:02
that the selected uh the ball selected from earn2 was read after we made this transfer and now i'm saying let's suppose that we know that the ball selected was red what was the probability now that the transferred ball was green okay so now let's go back to the same notation and and and information that i had before uh from from the first problem from the law total probability problem this is what i had this is the information that i had before but now again i'm going to
5:32
assume that the ball selected from r2 is red so i'm given that and i'm asked to calculate this probability so they're not telling you it's a conditional probability but it's actually it is a conditional probability i've listed i've written it down there in the bottom left i'm asked to calculate here the probability that the transferred ball is green given that the ball given that the ball selected uh from urn two was red and again that that's that's obviously a a conditional probability
6:01
i want to highlight something to you here if i turn if i turn those uh so i highlighted in red the the the two probabilities that i want to focus on if you look at the bottom left probability and compare that to the probability on the bottom right there i just interchange the two events and that's that's that's that's a bayes theorem problem so we're talking about a bay's theorem problem if you would interchange the two events it's an easy question to answer it's you
6:28
know what's the probability that the the color of the ball selected from r2 is red given that the transfer ball is green that's a that's an easy calcul that's an easy calculation that's an easy problem to answer but then when you switch the events and that's what bayes theorem does it switches the events uh and and then it becomes a little bit more difficult question okay so now let's look at the bottom left i want to i want to calculate this conditional probability so the first
6:54
thing i'm going to do is just use the facts with the fact about conditional probability that the probability of a given b is the probability of a and b divided by the probability of b and i'm going to apply that fact with a being the event that the transferred ball is green and b being the event that the color of the ball chosen from urn two was was red and so when i when i just apply these uh apply this this conditional probability fact this is the equation that i that i
7:21
get and now let's look at the numerator for just a second uh this is a probability of an a intersect b uh a being the transfer ball equals is is green and b being the color of the selected ball from r2 is red and now what i want to do is just interchange the rolls of a and b so i'm just going to swap them around so again all i did is swap the uh the events around so the probability of a and b is the same as the probability of b and a so i just switched the events around and now i'm
7:51
going to apply the the the fact that the probability of a intersect b is generally this conditional probability of a given b and then times the probability of b using a being with a being the event that the that the uh uh boss uh selected from earned two is red and b being the event that the transfer ball is green so again i switch the events around and when i do that and i apply uh apply this probability fact of a intersect b then this is the equation that i would get
8:24
all right and so look again i just kind of want to point out the fact i've i've started with a probability uh that's a conditional probability on the left bottom left that's kind of difficult to do and now what have i done i've rewritten it such that now i'm i'm using the conditional probability with the events reversed which was the easier calculation to do and that's base that's a base theorem that's what that's how you're going to do a base theorem problem every time
8:51
in order to get some more room i'm going to uh in that last line there i'm going to delete the middle uh the middle expression and just to get some more room here and now i want to uh now let's let's let's look at at each of these um well actually let's just look at the denominator the denominator is what we did before this i mentioned before that bayes theorem and the law of total probability are very related and this is where uh in the denominator when you
9:18
when you write base their amount in the denominator you're going to get a probability that you're going to need to calculate using the law of total probability and we just did that in the last example and so in the last example we saw that the denominator was the 0.25 times 0.6 plus 0.2 times times 0.4 now in the denominator when you use that law of total probability again this is how bayes theorem is going to work every time in that denominator you're going to have a
9:42
a sum of a couple a sum of terms in this case i have two terms i'm going to highlight the first term in red and the second term in in green so in the denominator when you when you rewrite uh the when you get the expression for basis uh for this probability and and and uh in the denominator we're going to use the law of total probability you're going to get an expression that has several terms and my point here is that the numerator will be one of those two
10:09
terms so now you just have to figure out which one of those two terms is the numerator going to be and of course the numerator corresponds to the uh transfer ball being green so i'm um i would i would replace the numerator with its values which is the the green values in the denominator okay so that's going to be uh that's how you're going to do bayes theorems that's they're all done the same way uh if we went through let's go through the rest of the calculation
10:36
in the numerator we're gonna get point zero eight the denominator we're gonna get point two three which you can multiply numerator and denominator by a hundred and and just write that as an 8 over 23. okay so uh again that's that's a typical base theorem problem uh it's going to use the law of total probability you're going to have to use the law of total probability in the denominator and then the denominator becomes a certain uh a sum of a a number of terms in this case
11:02
there were two terms there might be three or four terms in the denominator but then the numerator will be always be one of those terms in the denominator okay so uh that's it with this example i'll see you in the next video [Music]
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