Statistics and probability
Think about what probabilities really mean. What does a probability of 0 mean? How about 1?
- [Voiceover] What I hope to do in this video is give ourselves a more intuitive sense of probabilities. Let's go back to an example that we've seen before. We're rolling a fair six-sided die. There are six possibilities. We could get a one, a two, a three, a four, a five or a six. Now let's say we ask ourselves what is the probability of rolling a number that is less than or equal to two? What is this going to be? Well, there are six equally likely possibilities. Rolling less than or equal to two, well, that means I'm either rolling a one or a two. So two, one, two, of the six equally likely possibilities meet my constraints. So there is a 2/6 probability of rolling a number less than or equal to two. Or I can just rewrite that as an equivalent fraction. I could say there's a 1/3 probability. I could go either way. Now let's also ask ourselves another question. What is the probability of rolling a number greater than or equal to three? Once again, there are six equally likely possibilities. How many of them involve rolling greater than or equal to three? Let's see, one, two, three, four, these possibilities right over here. Throw a three, a four, a five or six. So four out of the six equally likely possibilities. Or I could rewrite this as an equivalent fraction, as 2/3. So what's more likely? Rolling a number that's less than or equal to two? Or rolling a number that's greater than or equal to three? Well, you can see it right over here. The probability of rolling greater than or equal to three is 2/3 while the probability of rolling less than or equal to two is only 1/3. This number is greater. So this has a greater probability or another way of thinking about it, rolling greater than or equal to three is more likely than rolling less than or equal to two. In fact, not only is it more likely, you see that 2/3 is twice 1/3. This right over here is twice as likely. You're twice as likely to roll a number greater than or equal to three than you are to roll a number less than or equal to two. You can even see right over here. You have twice as many possibilities of the six equally likely ones, four versus two. Four versus two here. So you say, "Okay, I get it Sal." If the probability is a larger number, the event is more likely. It makes sense and in this case, it's twice. The number is twice as large so it's twice as likely. But what's the range of possible probabilities? How low can a probability get and how high can a probability get? Let's think about the first question. How low can a probability go? How low, so what's the lowest probability that you can imagine for anything? Well, let's give ourselves a little bit of an experiment. Let's ask ourselves the probability of rolling a seven. Well, once and pause the video and try to figure it out on your own. Well, there are six equally likely possibilities. How many of them involve rolling a seven? Well, none of them. It's impossible to roll a seven. So none of the six. We could say this probability is zero. If you see a probability of zero, someone says the probability of that thing happening is zero, that means it's impossible. That means in no world can that happen, if it's exactly zero. This right here, the probability is zero. That means it is impossible. It is impossible. Now how high can a probability get? How high can a probability get? Well, let's think about it. Let's say probability of rolling any number from one to six. Well, I have six equally likely possibilities and any one of those six meets this constraint. I would have rolled a number, any number, from one to six, including one and six. So there are six equally likely possibilities. So the probability is one. Someone says the probability is zero, it's impossible. If someone says the probability is one, that means it's definitely going to happen. It's definitely going to happen. So the maximum probability for anything is one. The minimum probability is zero. You don't have negative probabilities and you don't have probabilities greater than one. You might be thinking, "Wait, wait. "I've seen things that they look like "larger numbers than one." You're probably thinking of seeing this as a percentage. One as a percentage, you can also write this as 100%. This right over here as a percentage is 100%. 100% is the same thing as one. You can't have a probability at 110%. 110% would be the same thing as 1.1. Now this is really interesting because you'd often see someone say, "Hey, something for sure is going to happen "or something is impossible." But even a lot of the things that we think for sure are going to happen, there's some probability or some chance that they don't happen. For example, you might hear someone say, "Well, what's the probability that the sun "will rise tomorrow?" Well, you might say it's going to happen for sure. But you gotta remember some type of weird cosmological event might occur, some kind of strange, huge planet-sized object in space might come and knock the earth out of its rotation. Who knows what could happen? All these have a very low likelihood. Very, very, very, very, very, very low likelihood. But it's hard to say it's exactly one. If I had said the probability that the sun will rise tomorrow, instead of saying one, I would probably say it's 0.999. I would throw a lot of nines over here. I wouldn't say it's 0.9 repeating forever. Actually, there's an interesting proof that 0.9 repeating forever is actually the same thing as one, which is a little counterintuitive. But I would say there's a very high probability. But even if it's such a high probability, it's going to be close to one. But I won't say it's exactly the one because there could be some kind of quasar that blasts us with gamma rays. Or who knows what might happen? But it's a very, very high probability. Same thing, the probability here, probability that my pet gopher could write the next great novel. Writes a novel. Actually not just a novel, a great novel. Just a novel wouldn't be that impressive for a gopher. Let's say great novel. Well, once again, this gopher sitting there typing at a keyboard. It would seem somewhat random but there is some probability that it actually does it. There's some chance it does it. So I would put this at a very low and I wouldn't say it's exactly zero. If we had an infinite number of gophers doing this forever, who knows, maybe one of them might write that great novel. In fact, if we had an infinite number doing it forever, eventually, a lot of people would say, at some point you would. But just one gopher trying to write a novel, what's the probability they write a great novel? I would say it's pretty close to zero. I'd throw a lot of zeros here, and at some point you might have something like this. Once again, not absolutely impossible but pretty close to, pretty, pretty close to impossible. So big takeaways? Higher probability, more likely. The lowest probability you can get to? Zero. Highest probability is one. If your probability is more, when you're talking about coin flipping. If you say the probability of heads for a fair coin and you say, "Well, that's 1/2," that means it's equally likely to happen or not happen. Anything that has a larger probability than 1/2, it's more likely to happen than not. Anything that has a probability of less than 1/2, it's less likely to happen than not.