Bayes' Theorem of Probability With Tree Diagrams & Venn Diagrams

In this video we will focus on Bayes'

theorem

, but first let's review some formulas, particularly with conditional

probability

, the

probability

that event a will occur, given that event b has already occurred, is the probability that events a and b will occur divided by probability. that event b will occur in the same way we can say that the probability of b given a is equal to the probability that b and a occur divided by the probability that event a occurs now the probability that event a and b occur is equal to the probability that by a event will occur, so basically what I'm doing is setting these two equal to each other.

Now this is equal to the probability of a given b multiplied by the probability of b, so let's replace that on the left side now. On the right side, the probability of b and a occurring is the product of the probability of b occurring given that a has already occurred times the probability of a occurring. Now what I'm going to do is divide both sides by the probability of a occurring. event b will occur and this will give me the formula associated with Bayes'

theorem

, so the probability that a will occur given that b has already occurred is equal to the probability that b will occur given that a has occurred multiplied by the probability of the event a occurs divided by the probability of event b occurring, so Bayes theorem helps us calculate the conditional probability of an event if you basically know the inverse conditional probability with some other probability values ​​as well, but this is the basic formula of the base term so p of b given is equal to p of b given a multiplied by p of a divided by p of b now let's use a simple example that will illustrate the use of the base theorem so let's say that in a bottle There are a lot of pieces of paper. with numbers let's say that event a represents the following numbers that are drawn one two three four and five now let's say that event b represents the numbers four five six seven eight and nine now what is the probability of a given b let's use Bayes' theorem To calculate this particular probability, although it is not necessary for educational purposes, let's do it now.

What I'm going to do is create a Venn diagram, so let's say the first circle is for event a and the second is for event b, so a and b have these two numbers in common four and five so that's the intersection of a and b a also has the numbers one two and three b has the number six seven eight and nine now to calculate this using the base theorem we need to know the probability of event a occurs the probability that event b occurs and basically the inverse conditional probability the probability that b occurs given that a has already occurred so what is the probability that a occurs?

We need to write the sample space the sample space represents all of the possible numbers, the numbers in a and b, so the numbers we can get represent all the natural numbers from 1 to 9. so a has five of the nine possible numbers between which we can choose, so the probability that event a occurs assuming that each number has the same probability of being drawn from the bottle, will be 5 out of 9. Now b has six numbers out of the nine numbers here, so that the probability of event b occurring is now six out of nine. What about the next one?

What is the probability that b occurs given that a has already occurred? There's a formula for what we talked about earlier in this video, but to think about it conceptually, how much b is there in a? numbers one to five of those five numbers there are only two numbers that are in b that are part of a, so the probability of b occurring given that event a occurred is two out of five, so now with this information we can calculate this. particular conditional probability the probability that a occurs given that b has already occurred, it will be p of b of a or rather b given a multiplied by p of a over p of b and let me get rid of this, so p of b given a that's what we have here, which is 2 over 5 times p of a, which is 5 over 9 divided by p of b, which is 6 over 9.

So now we could cancel out the five and if we multiply the top and bottom of the fraction by nine We can also cancel out the nines, so we are left with two over six, which reduces to one over three, so the probability of a occurring given that b has already occurred is one in three. Now let us confirm that using conditional probability, then p of a given b is the probability that events a and b occur over the probability that b occurs, so we know that the sample space has nine numbers, how much of that is a and b, the intersection of a and b is basically two numbers four and five. so the probability of events a and b occurring is two of the nine possible numbers now the probability of b occurring is what we see here is six over nine, so multiplying the upper and lower numbers by nine we will obtain the same result initially It will be two out of six, which reduces to one out of three.

Now, we could have done this from the beginning, but now you see how Base Derm works with the formula that we wrote earlier, but let's consider another example where we could use the Bayes theorem formula now for those of you who really value this video . If you want to show your appreciation, one of the best ways to do it is by subscribing to this channel and it really doesn't take much time, just click that red button at the bottom of the screen and that's it, by the way, if you decide to do it , don't forget to turn on or click that notification bell now for those of you who want to support my channel, here is the link to my Patreon page, also when you get a chance check out the links in the description section below this video because I'm going to post some other resources that you may find useful, so let's get back to the video, one study in particular showed that 12 of men are likely to develop prostate cancer at some point in their lives.

A man with prostate cancer has a 95 chance of receiving a positive result on a medical test. A man without prostate cancer has a six percent chance of getting a false positive result. What is the probability? that a man has cancer given that he has a positive test result, go ahead and try this problem, feel free to take a minute if you need to, so let's write down what we know about the probability that a man has cancer. I'm going to write pfc. is 12, which is 0.12 as a decimal. Now the second sentence says that a man with prostate cancer has a 95 percent chance of getting a positive test result, so given that the person has cancer, the probability of him getting a positive result or a positive test is 95 or 0.95 now what about the third sentence?

A man without prostate cancer has a six percent chance of getting a false positive result, so the probability of getting a positive result given that the person does not have prostate cancer is six percent or 0.06, so What is the probability that the person has cancer given that he or she has a positive test result? That's what we need to find, so let's write the equation for that using the Bayes term, so this will be the inverse conditional probability, which is the probability of getting a cancer. positive test result given that the person has cancer multiplied by the probability that the person has cancer divided by the probability that the person has a positive test result now we already have these two values ​​but we don't have this one, how can we find it in In this case, it will be very useful to make a

tree

diagram so that 12 of the population have prostate cancer according to this particular study, which means that 88, which is 100 minus 12 or 0.88, will not have cancer now among those who have cancer. of them will have a positive result, which means the other five percent will have a negative result.

Now, of those who do not have cancer, six percent will test positive, which means the other 94 percent will test negative. Using this

tree

diagram, how can we determine the probability that a person will have a positive test result? The probability that a person will have a positive test result depends on two events: the probability that the person will have cancer and have a positive test. result or which will be represented by the plus symbol the probability that the person does not have cancer and has a positive result those are the two options then the probability that the person has cancer and has a positive result is the product of these two values where you will get cancer and have a positive test result, so it's basically 0.12 multiplied by 0.95.

The probability that the person does not have cancer and has a positive test result is the product of these two values, so it is point 88 multiplied by zero six point twelve. point nine five plus point eighty-eight times point zero six that's equal to point one six six eight now I'm running out of space so I'm going to have to delete some things, I hope you wrote it down because I'm going to refer to If not , you can always rewind the video. Now let's plug the numbers into this formula so that the probability of getting a positive test given that the person has cancer is 0.95, the probability that the person has prostate cancer is 12 percent, and the probability that men or amen will have a positive test result will be 0.1668 or 16.68 percent, so it's 0.95 times 0.12 divided by 0.1668, so you should get 0.68345, so there's about a 68.3 chance that the person has cancer given that they have a positive result. test result, then what is the probability that the person does not have cancer even though they have a positive result?

It's going to be one hundred minus this number, which is 100 percent minus 68.3 or 1 minus 0.683, which is 0.317, so there's about 31.7 percent. possibility that even if the person has a positive test result, they do not have cancer, so sometimes even if they have a positive test result, it does not mean that they have that condition, that there is always a risk factor, but the answer we are looking for. because in this problem there is the number 68.3. Now there is another way to get that answer. Let's talk about how we can do it. Let's say there are 10,000 people in a city, and according to a survey, 12 percent of those people have cancer. so we are going to use the same percentages 10,000 multiplied by 0.12 is 1,200 and the other 88 do not have cancer, so 10,000 multiplied by 0.88 is 8,800.

Now, of those who have cancer, we know that 95 percent of them will get a positive test result, so 95 out of 1,200 or 1,200 multiplied by 0.95, that's 1,140, ​​so the other five percent, which is one thousand two hundred times point zero five, will get a negative test result. of the test, so now it's sixty of those who don't have cancer, we know six percent of that. The population will get a positive test result, so 8800 times 0.06, that's 528, the other 94 percent will get a negative test result when they don't have cancer, so 8800 times 0.94, that's 8272. So Let's now see if we can use this information.

To get the same answer, let me clarify some things so that the probability of a person having cancer, given that the person has a positive test result, is good, let's divide it into two parts according to the number of people we choose, it doesn't matter If you choose ten thousand or one hundred thousand because we are dealing with a proportion, it will still be the same of this group of 10,000 people, how many have a positive test result, the number of people who have a positive test result is the sum of those two numbers is 1140 plus 528 now, of those who have a positive test result, how many of them have cancer?

This number alone is the number of people who have cancer and test positive, so that's 11 40. so let's go ahead and add 1140 plus 528, which gives 16 68, so 1140 divided by 1668 gives us 0.683 so we can see why the answer is the same, there is a 68.3 probability that the person has cancer given that they test positive. test result, you now see two ways you can get the answer using a tree diagram or using the formula associated with bass therm thanks for watching