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Studying for a test? Prepare with these 8 lessons on Probability and combinatorics.
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So far, we've been dealing with one way of thinking about probability, and that was the probability of A occurring is the number of events that satisfy A over all of the equally likely events. And this is all of the equally likely events. And so in the case of a fair coin, the probability of heads-- well, it's a fair coin. So there's two equally likely events, and we're saying one of them satisfies being heads. So there's a 1/2 chance of you having a heads. The same thing for tails. If you took a die, and you said the probability of getting an even number when you roll the die. Well, there's six equally likely events, and there's three even numbers you could get. You could get 2, a 4, or a 6. So there's three even numbers. So once again, you have a 1/2 chance of that happening. And this is a really good model where you have equally likely events happening. Now I'm going to change things up a little bit. So I'm going to draw a line here because this was just one way of thinking about probability. Now we're going to introduce another one that's more helpful when we can't think about equally likely events. And in particular, I'm going to set up an unfair coin. So this right over here is going to be my unfair coin. So that is my coin. Well, I could draw the coin. So it's a gold coin this time. It is unfair. One side of that coin is a little heavier than the other, even though it's meant to look fair. So it still has that picture of some president or something on one side of it. So this is the head side. This is heads, and then, obviously, on the back, you have tails. But as I mentioned, this is an unfair coin. And I'm going to make it interesting statement about this unfair coin and one that really doesn't fit into the mold that I set up over here, and this interesting statement is that we have more than a 50/50 chance of getting heads or more than a 50% chance or more than a 1/2 chance of getting heads. I'm going to say that the probability of getting heads for this coin right over here is 60%. Or another way to say it, it's 0.6. Or another way to say it, it is 6 out of 10. Or another way to say it, it is 3/5. And this might make intuitive sense to you and hopefully it does a little bit, but I want you to realize that this is fundamentally different than what we were saying before because now we can't say that there are two equally likely events. There are two possible events. You can either get heads or tails. We're assuming that the coin won't fall on its edge. That's impossible. So you're either going to get heads or tails, but they're not equally likely anymore. So we really can't do this kind of counting the number of events that satisfy something over all of the possible events. In this situation, in order to visualize the probability, we have to kind of take what's called a "frequentist approach" or think about it in terms of frequency probability. And the way to conceptualize a 60% of getting heads is to think, if we had a super large number of trials, if we were to just flip this coin a gazillion times, we would expect that 60% of those would come up heads. It's unclear how I determined that this is 60%. Maybe I ran a computer simulation. Maybe I know exactly all of the physics of this, and I could completely model how it's going to fall every time. Or maybe I've actually just run a ton of trials. I've flipped the coin a million times, and I said, wow, 60% of those, 600,000 of those, came up heads. And then, we could make a similar statement about tails. So if the probability of heads is 60%, the probability of tails-- well, there's only two possibilities, heads or tails. So if I say the probability of heads or tails, it's going to be equal to 1 because you're going to get one of those two things. You have 100% chance of getting a heads or a tails, and these are mutually exclusive events. You can't have both of them. The probability of tails is going to be 100% minus the probability of getting heads, and this, of course, is 60%. So it's 100% minus 60%, or 40%, or as a decimal, 0.4, or as a fraction, 4/10, or as a simplified fraction, 2/5. So, once again, this probability is saying-- we can't say equally likely events. We could say that, if we're going to do a gazillion of these, we would expect, as we get more and more and more trials, more and more flips, 40% of those would be heads. Now, with that out of the way, let's actually do some problems with this. So let's think about the probability of getting heads on our first flip and heads on our second flip. So, once again, these are independent events. The coin has no memory. Regardless of what I got on the first flip, I have an equal chance of getting heads on the second flip. It doesn't matter if I got heads or tails on the first. So this is the probability of heads on the first flip times the probability of heads on the second flip, and we already know. The probability of heads on any flip is going to be 60%. I'll write it as a decimal. It makes the math a little bit easier, 0.6, 0.6, and we can just multiply. I'll do it right over here. So this is 0.6 times 0.6. Now, it's always good to do a reality check. One way to think about it is I'm taking 6/10 of 6/10, so it should be a little bit more than half of 6/10 or probably a little bit more than 3/10. And we've explain this in detail where we talk about multiplying decimals, but we essentially just multiply the numbers, not thinking about the decimals at first. 6 times 6 is 36. And then you count the number of digits we have to the right of the decimal. We have one, two to the right of the decimal. So we're going to have two to the right of the decimal in our answer. So it is 0.36, and that makes sense. We're taking 60% of 0.6. We're taking 0.6 of 0.6, a little bit more than half of 0.6. And, once again, it's a little bit more than 0.3. So this also makes sense. So it's 0.36. Or another way to think about it is there's a 36% probability that we get two heads in a row, given this unfair coin. Remember, if it was a fair coin, it would be 1/2 times 1/2, which is 1/4, which is 25%, and it makes sense that this is more than that. Now, let's think about a slightly more complicated example. Let's say the probability of getting a tails on the first flip, getting a heads on the second flip, and then getting a tails-- I'm going to do this in a new color-- and then getting a tails on the third flip. So this is going to be equal to the probability of getting a tails on the first flip because these are all independent events. If you know that you had a tail on the first flip, that doesn't affect the probability of getting a heads on the second flip. So times the probability of getting a heads on the second flip, and then that's times the probability of getting a tails on the third flip. And the probability of getting a tails on any flip we know is 0.4. The probability of getting a heads on any flip is 0.6, and then the probability of getting tails on any flip is 0.4. And so, once again, we can just multiply these. So 0.4 times 0.6. There's actually a couple of ways we can think about it. Well, we could literally say, look, we're multiplying 4 times 6 times 4, and then we have three numbers behind the decimal point. So let's do it that way. 4 times 6 is 24. 24 times 4 is 96. So we write a 96, but remember, we have three numbers behind the decimal point. So it's one to the right of the decimal there, one to the right of the decimal there, one to the right of decimal there. So three to the right. So we need three to the right of the decimal in our answer. So one, two-- we need one more to the right of the decimal. So our answer is 0.096. Or another way to think about it is-- write an equal sign here-- this is equal to a 9.6% chance. So there's a little bit less than 10% chance, or a little bit less than 1 in 10 chance, of, when we flip this coin three times, us getting exactly a tails on the first flip, a heads on the second flip, and a tails on the third flip.