Conditional Probability (1 of 7: A surprising example)

[salamiiscrazi]

If I had this kind of teacher, maybe it's just me, but I would try to never be late to class. He makes it too interesting and stimulating for me to come late and mis out. Honestly great teaching

[superdupernice]

For anyone that wants to know how to calculate this.

For conditional probability, the equation P(A n B)/P(B) can be used to find the probability of A given B. In our problem, we're looking for the probability of being sick given that a person tested positive. For our problem, this can be written as P(Sick n Correct)/P(Positive). You can also write down P(Sick n Positive)/P(Positive), but I'm writing down P(Sick n Correct) instead in order to distinguish a difference between a positive result and a correct result.

First, lets find P(Sick n Correct). For this problem, we assume that the accuracy of the test is independent of the health of the patient. This basically means that being sick or healthy doesn't change the accuracy of the test being correct of incorrect. Since we're assuming independence, P(Sick n Correct) will just be the probability of being sick multiplied by the probability of getting the correct diagnosis (correct diagnosis meaning positive for a sick person). In this case, we get (.001)*(.95)=.00095.

For the second part of the problem, we want to find the probability of getting a positive result. There are two possible ways of getting a positive. Either you're sick and got the correct diagnosis, or you're healthy and got the wrong diagnosis. Mathematically, we can write this down as, P(Positive) = P(Sick n Correct)+P(Healthy n Wrong). This would be (.001)*(.95)+(.999)*(.05)= 0.0509.

Plugging the numbers into the equation from earlier, we get 0.00095/0.0509=0.01866=1.866%, which is roughly 2%.

If you want to solve this by using expected values, in a sample of 10000 people, we would expect 509 people to test as positive. This number comes from using P(Positive)=.0509, then multiplying it by 10000. Of this 10000, we would also expect 9.5 of these positives to also be true positives.The number comes from P(Sick n Correct)*10000. By doing 9.5/509, you get the same 1.866% from above.

[paulolameiras]

I have seen several videos of this teacher. He normally does a good job explaining. In this particular video I can't help feeling that something went wrong. If the test has a 95% accuracy of being correct, the obvious answer seems to be 95%. But on the other hand we know that we only have a 0.1% chance of having the disease. There is a false zone of 5% and a true zone of 95% regarding the test results. The only factor unknown is, in which of these zones is the person with the disease going to fall, so we should consider two scenarios:
1)- The person with the disease falls in the false zone (5%)
2) - The person with the disease falls in the true zone (95%)
If we consider a sample of 1000 people, in the first scenario we have 950 true negatives + 49 false positives + 1 false negative.
In the second scenario we have 949 true negatives + 1 true positive + 50 false positives.
Since the patient had a positive result we are only interested in finding the chance of true positives against all the positives (being true or false). In the first scenario we don't have any true positives so the probability of having the disease is 0%. In the second scenario we have 1 true positive and 50 false positives, so the probability of having a positive test and actually having the disease (true positive) should be 0.95 * (1/51)= 0.0186274, meaning approximately 1.86 %. Notice that Mr. Woo considered a different scenario, he actually divided (1/50), meaning that he considered the person with the disease to be in the false zone, which would make that person a false negative. Since the patient was diagnosed positive, the scenario presented in this video would be impossible to occur. Overall, the logic is correct, it just failed in the details and the precision of the calculation.

[jacopostortini]

I thought that 95% accurate meant that on a hundred of people making the test, 95 get the right answer and 5 get the wrong answer. So when you get positive result you have the 95% of probability that it's right

[Sunset_vibes_]

I have a Proabilities exam in 20 days
You just saved my life
Thank

[toituxu]

Best lesson from motivational, full of energy teacher ever!

[gedlangosz]

Bayes Law.
P(A|B) = P(B|A).P(A) / P(B)
So the probability of having the disease given a +ve result is
0.95 X 0.001 / (0.95 X 0.001 + 0.05 X 0.999)
Which is about 1.866%. The example on the board has missed out false negative results.

I recently witnessed this for real. A friend was told they tested positive for a disease even though the doctor thought this was unexpected. A retest with another lab was recommended which came back negative. My friend was really upset and worried waiting for the second set of results. My friend is not mathematical in the least and me explaining Bayes Law didn't really help. The doctor seemed ignorant of it too.

[renanruseler]

The video is very good. I calculated everything in my notebook and the probability is aproximately 1, 8664047%. You just have to divide 0, 095% (probability of a true positive) by 5.09% (the probability of a positive, whether the result is accurate or not). He just showed the resolution in a simpler way.

[linzhou]

2% is not the precise answer. The reason is very simple: there are 2 possibilities that the test result is positive: 1. You are indeed sick, assume the probability of it is P1 2. The test result is wrong, assume tbe probability of it is P2. So the probability that you are indeed sick if the test result is positive is equal to

[lm]

The teacher mentioned symptoms and a raised suspicion in a doctor prior to doing a test. That means the pre-test probability was no longer at the low level of 0.1% background prevalence.

[EddieYu]

That is why when describing results of diagnostic tests, terminologies such as sensitivity, specificity, false positive rate and false negative rate are used rather than "accuracy". "Accuracy" is a valid scientific term but understanding what's really happening isn't very intuitive. When describing a test result to a patient, former 4 parameters are mainly used rather than accuracy because by definition, those 4 have more straightforward meanings.

[趙佶-bq]

5% does not mean the 5% test results has to be “wrong”, it means the test result is not reliable.

Otherwise that one person can’t be in the red quadrant since the red quadrant is “all wrong “, because that person is being tested positive where he is actually positive, which means the test result is correct and that means it lies into the 95%

[RM-fsub]

Was so happy, I wasn't there to shine in ignorance before the students probably less than half my age! Used to read about specificity and sensitivity just towards the exams for safety's sake hoping my short term memory fuse would not blow and hopefully keep it handy for a while before it ended up in my brain's trash bin! Nassim N Taleb caught the medical field people in this similar predicament on probabilities, percentages - and most famously - the P value.

[Lurker]

Somehow the first run-through of the video went over me pretty quickly and I thought I understood it. However, on my second run-through, I just couldn't wrap my head around the meaning behind the word "accurate."


So following Mr. Woo's narrative with the single individual being tested positive for the disease within a population of 1, 000 people, would it be reasonable to assume that of the 1, 000 people 50 of them had "inaccurate" test results including both false-positive and false-negative results? Since you were tested positive, would that mean the 50 people with "inaccurate" results all be apart of the false-positive group leading to the 1/50 chance of you actually having the disease? If so, then what happened to the false-negative group?


Perhaps test inaccuracy also applies to true-positives and true-negatives and I'm just looking at this problem at the wrong angle.

[nikhil__]

Please make videos on bayes' theorm I have understood the conditional probability but how it works in bayes' theorm I don't get that so please make a video on it!!!

[c-math]

Please we need more videos about probability

[mamatheshkumar]

probability and graph matrix set theory.

[richdewald]

This assumes that the doctor has no idea of who actually has the disease from the symptoms. If the probability of having the disease from the symptoms diagnosed by the doctor was 0.1% was correct, then 2% is correct. The probability of the person having the disease is also a function of the doctor's ability to predict he is correct from the symptoms.

[zero]

Maybe I'm not sure but, somehow, I actually reckon that the person who is mentioned in this problem is out of the red box. Because:
Case 1: If this person is out of red box ( have correct result ), he ( this person ) will have a positive test. In addition, all the people in the red box will have the same one => probability is 1/51
Case 2: If this person is in the red box ( the people who have inaccurate result ), he obviously have the NEGATIVE test. Becasue, in fact, he have a disease and the only one who have a disease in 1000 people.
Is it right ??? Hoping for more explanations, please !!!!
Thank you^^

[gigz]

I hate "distractors" on multiple choice tests. In order to get the question right, you first have to pass the test of not getting "distracted" before allowing yourself enough time to fully think about the problem. There is a confirmation trap in seeing the exact answer that you calculated appear in the answer options. While possibly a good life skill, it dilutes the effectiveness of the question at assessing whether or not the student knows the material/correct answer. Ironically, conditionally probability could be applied here.