Conditional Probability

And once again, we’d like to refer the corresponding numbers over here on the right side 0.1 for the cancer times the probability of getting a negative result conditioned on having cancer and that is 0.1 0.1, which is 0.01.

Moving to the next case–what do you think the probability is that the person does have cancer but the test comes back negative? What’s the combined probability of these two cases?

And the answer is probability of cancer is 0.1, probability of test being positive given that he has cancer is the one over here–0.9, multiplying those two together gives us 0.09.

Look at this, this is very nontrivial but armed with this, we can now build up the truth table for all the cases of the two different variables, cancer and non-cancer and positive and negative tests outcome. So, let me write down cancer and test and let me go through different possibilities. We could have … Read more

And the answer is 0.8. As I’m sure you noticed in the case where there is cancer, the possible test outcomes add up to 1. In the where there isn’t cancer, the possible test outcomes add up to 1. So 1 – 0.2 = 0.8.

Of course, in reality, we don’t know whether a person suffers cancer, but we can run a test like a blood test. The outcome of it blood test may be positive or negative, but like any good test, it tells me something about the thing I really care about–whether the person has cancer or not. … Read more

Of course, in reality, we don’t know whether a person suffers cancer, but we can run a test like a blood test. The outcome of it blood test may be positive or negative, but like any good test, it tells me something about the thing I really care about–whether the person has cancer or not. … Read more

And the answer is 0.9 with just 1 minus the cancer.

So there’re important lessons in what we just learned, the key thing is we talked about conditional probabilities. We said that the outcome in a variable, like a test is actually not like the random coin flip but it depends on something else, like a disease. When we looked at this, we were able to … Read more

And the answer is depressing. If you, once again, draw the truth table, you find, for the different combinations, that if you’ve drawn coin 1, you’d never see tails. So this case over here, which indeed has tails, tails. We have 0 probability. We can work this out probability of drawing the first coin at … Read more

Let me do this once again. There are 2 coins in the bag, coin 1 and coin 2. And as before, taking coin 1 at 0.5 probability. But now I’m telling you that coin 1 is loaded, so give you heads with probability of 1. Think of it as a coin that only has heads. … Read more

This is a non-trivial question, and the right way to do this is to go through the truth table, which I’ve drawn over here. There’s 3 different things happening. We’ve taken initial pick of the coin, which can take coin 1 or coin 2 with equal probability, and then you go flip it for the … Read more

Now let me up the ante by flipping this coin twice. Once again, I’m drawing a coin from this bag, and I pick one at 50% chance. I don’t know which one I have picked. It might be fair or loaded. And in flipping it twice, I get first heads, and then tails. What’s the … Read more

And let’s do the truth table. You have a pick event followed by a flip event We can pick coin 1 or coin 2. There is a 0.5 chance for each of the coins. Then we can flip and get heads or tails for the coin we’ve chosen. Now what are the probabilities? I’d argue … Read more

So now what happens is, I’m going to remove one of the coins from this bag, and each coin, coin 1 or coin 2, is being picked with equal probability. Let me now flip that coin once, and I want you to tell me, what’s the probability that this coin which could be 50% chance … Read more

And the answer is 0.5 for coin 1and 0.1 for coin 2, because these things have to add up to 1 for each of the coins.

To do so, let’s study a medical example–supposed there’s a patient in the hospital who might suffer from a medical condition like cancer. Let’s say the probability of having this cancer is 0.1. That means you can tell me what’s the probability of being cancer free.

This time around, we have a bag, and in the bag are 2 coins,coin 1 and coin 2. And in advance, we know that coin 1 is fair. So P of coin 1 of coming up heads is 0.5 whereas coin 2 is loaded, that is, P of coin 2 coming up heads is 0.9. … Read more

Putting all of this into mathematical notation we’ve given the probability of having cancer and from there, it follows the probability of not having cancer. And they give me 2 conditional probability that are the test being positive. If we have have cancer, from which we can now predict the probability of the test being … Read more

And the result, once again, is found in the truth table, which is why this table is so powerful. Let’s look at where in the truth table we get a positive test result. I would say it is right here, right here. If you take corresponding probabilities of 0.09 and 0.18, and add them up, … Read more