3-4-4. Probability

And the answer is 1. To see this, we know that the probability of tails is 0. All the probability goes to heads. 1 1 = 1 1 0 = 0 0 1 = 0 And 0 0 = 0. And it’s easy to verify that all these things add up to 1. Our (H,H) … Read more

Let’s now go to the extreme, and this is a challenging probability question. Suppose the probability of heads is 1, so my coin always comes up with heads. What is the probability of (H,H)?

And, not surprisingly, it’s 1. That is, the truth table always has a probability that adds up to 1 because it considers all possible cases, and all possible cases together have a probability of 1. So we just check this and make sure it’s correct. Reading from this table, we find that the probability of … Read more

I have here a U.S. dollar coin. It has two sides, one showing a head and one showing what’s called tails. In probability, I’m giving a description of this coin, and I’m making data. We just make data. [sound of coin spinning] So if we look at the coin, it came up heads. So I … Read more

If you add up these numbers over here– please go ahead and add them up and tell me what the sum of those numbers is.

And the answer using our product rule is heads, heads comes out to 0.6 0.6, which is 0.36. Heads followed by tails is 0.6 0.4, which is 0.24. Tails followed by heads is, again, 0.24. And tails followed by tails is 0.16, which is 0.4 0.4.

And now please fill out the entire truth table. There are four values over here, so please compute them for me.

And the answer is 0.4 because heads comes up 0.6, and 1 – 0.6 = 0.4.

Let me now challenge you and give you a loaded coin I flipped twice. And for this loaded coin, I assumed the probability of heads is 0.6. That really changes all of the numbers in the table so far, but you can apply the same method of truth tables to arrive at an answer for … Read more

That was a tricky question, and you couldn’t really know the answer if you’ve never seen probability before, but the answer is 0.25. And I will derive it for you using something called a truth table. In a truth table, you draw out every possible outcome of the experiment that you conducted. There were two … Read more

In our example, we observed heads twice. So now I want to ask you a really tricky question: What’s the probability of observing heads and heads if you flip the same unbiased coin twice? This means in each flip we assume the probability of heads is 0.5. Please answer here.

So we just learned something important. There’s a probability for an outcome; I’m going to call it A, for now. And we learned that the probability of the opposite outcome, which we’re going to call ¬A (this over here just means “not”) is 1 minus the probability as expressed right over here. That’s a very … Read more

And the answer is 0.25, which is 1 – 0.75 using the law down here. As you can verify, 0.75 + 0.25 =1.

So suppose the probability of heads is 0.75, that is, 3 out of 4 times we’re going to get heads. What is the probability of tails?

Statistics and probability are different but strongly related fields of mathematics. In probability, we make predictions about future events based on models or causes that we assume whereas in statistics we analyze the data from the past events to infer what those models or causes could be. There’s almost an opposite relation between these two. … Read more