7-1. Probability

And the answer is, well there’s two ways that we can achieve this. One is the all heads, and one all tails. You already know that 4 times heads is 1/16. And we know that 4 times tail is also 1/16. These are completely independent events. So the probability of either one occurring is 1/16 … Read more

Now let’s flip the coin 4 times and let’s call xi the result of the i-th coin flip. So each xi is going to be drawn from heads or tails. What’s the probability that all four of those flips gives us the same results no matter what it is, assuming that each one of those … Read more

And the answer is 0.125. Each head is a probability of a half. We can multiply those probabilities because they are independent events. And that gives us 1/8 or 0.125.

Here’s another quiz. What’s the probability that the coin comes up heads, heads, heads, three times in a row? Assuming that each one of those has the probability of a half, and that these coin flips are independent.

And the answer is three quarter, it’s a loaded coin. And the reason is either one could come up with certain probability. The total of those is 1, the quarter is claimed by heads, therefore three quarters remain for tail, which is the answer over here.

Maybe after my next quiz. Suppose the probability of heads is quarter, 0.25. What’s the probability of tail?

So you just learned about what’s probably the most important piece of math with this task, in statistics, called Bayes Rule. It’s invented by Thomas Bayes, who was a British mathematician and a Presbyterian minister in the 18th century. Bayes rule is usually stated as follows. P(A|B) where B is the evidence and A is … Read more

So, the right answer is a half, 0.5, and the reason is the coin can only come up heads or tails, we know that it has to be either one. Therefore, the total probability of both coming up is 1. So, if half of the probability is assigned to heads, then the other half is … Read more

And the correct answer is 0.043. So, even though it received the positive test my probability of having cancer is just 4.3% which is not very much given that the test itself is quite sensitive. It gives me a 0.8 chance of Giving a negative result, if you don’t have cancer it gives me a … Read more

Now our next quiz, I want you to fill in the probability of cancer. Given that we just received a positive test.

And here the correct is 0.009 which is the product of the prior 0.01 times the conditional 0.9. Over here we get 0.001 probability of prior cancer times 0.1. Over here you get 0.198, probability of not having cancer is 0.99 times still getting a positive reading with is 0.2 and finally we get 0.792 … Read more

Now ultimately I’d like to go what’s the probability I have this cancer, given that I’ve just received a single positive test. Before I do this, please help me filing out some other probabilities that are actually important. Specifically, the joint probabilities. The probability of a positive test and having cancer. The probability of a … Read more

0.1 which is the difference between 1 and 0.9. Let’s assume the probability of the test coming out positive even if we don’t have this cancer is 0.2. In other words, probability if it is correctly saying you don’t have the cancer if you’re cancer free is 0.8.

Let’s assume there’s a test for this cancer, which I’ll give those probabilistically an answer whether we have this cancer or not. So let’s say, the probability of a test being positive is indicated by this plus sign, given that we have a cancer, is 0.9. Probability of the test coming out negative if we … Read more

1 And yes, the answer is 0.99.

Next example is a cancer example. Suppose there’s a specific type cancer which exists for 1% of the population. I’m going to write this as follows. You can probably tell me now what the probability of not having this cancer is.

So the correct answer here is 0.78 and over here it’s 0.756. To get there, let’s complete this one first, probability of D2 equals sunny. Well, we know there’s a 0.9 chance it’s sunny on D1 and then if it’s sunny, that it would stay sunny is a .0.8 chance. So we multiply these two … Read more

[BLANK_AUDIO] So let’s start with probabilities. Probabilities are the cornerstone of artificial intelligence, and they are used to express uncertainty. And the management of uncertainty is really key to many, many things in AI, such as machine learning and Bayes network inference, and filtering, and robotics, and computer vision and so on. So let’s start … Read more

A sunny day followed by rainy day is 0.6 chance and a rainy day follows a rainy day. Please give me a number.

Well, the correct answer is 0.2, which is the negation of this event over here.