Math Antics - Basic Probability

Hi, I'm Rob. Welcome to Math Antics. In this video, we're going to learn about how to do math with things that only sometimes happen. They might be likely or unlikely. We're going to learn about Probability. Usually in math, we deal with things that always happen the same way. They're completely certain. Like if you add 1 and 1, you're always going to get 2. If you multiply 2 and 3, you're always going to get 6. There's no uncertainty at all. But in the real world, things aren't always so predictable. Take a coin toss for example. We can't predict whether it'll be heads or tails. It's unpredictable or random. And that's why some people will flip a coin to help decide which of two things to do. That's how I make every decision in life. Why am I not surprised? Oh no! Good luck! But even though we don't know what each coin flip is going to be, we do know a few things about it. We know that with a fair coin toss, that heads is just as likely to show up as tails. The probability of an event, like getting heads or getting tails, is a value that tells us how likely that event is to happen. With our coin toss, since each side is just as likely and there's only two sides to a coin, if we flipped a coin a lot of times, we should expect that about half the flips will be heads and about half the flips will be tails. That means that the probability of flipping heads is the fraction one-half and the probability of flipping tails is also one-half. Let's look at this in a little more detail on something called a probability line. It's a number line that goes from 0 to 1. A probability of 0 means that an event CANNOT happen. It's impossible. And a probability of 1 means that an event is DEFINITELY going to happen. It's certain. That's why the probability line only goes from 0 to 1. An event can't be less likely than impossible, and it can't be more likely than certain. A probability of one-half, like with our coin toss, means that an event is just as likely to happen as it is to not happen. A probability less than one-half means that an event is unlikely, and a probability greater than one-half means that an event is likely. Oh, and in addition to fractions, it's also common to write probabilities as decimals or percentages since you can easily convert between those three. A probability of 0 is the same as a 0% chance of something happening. A probability of one-half is the same as a 50% chance of something happening. And a probability of 1 is the same as a 100% chance of something happening. Now that you know how a coin toss works, let's see an example of an event that is unlikely, using something a little more complicated than a coin. Let's take a look at dice. A standard die has 6 sides numbered 1 through 6. When you roll it, any of those sides is just as likely to come up as the others. That sounds a lot like flipping a coin, doesn't it? Each side of a die is just as likely to come up as the others, and each side of a coin was just as likely to come up as the other. So you might expect that the probability of rolling a 3 is 50%. But remember, with a coin toss, there are only two possibilities… heads or tails. With dice, there are 6 possibilities. That's going to make a difference in its probability. One way to think about it is that it's certain that one of those 6 sides will land facing upwards, which is a probability of 1 or 100%. But since only one side can face upwards for a given roll, we have to divide up that value among all the possibilities. In the case of a coin toss, since there were only two possibilities, we had to divide the probability by 2. 1 divided by 2 is ½, which is the decimal 0.5 or 50%. But with the die, we need to divide the probability up evenly between 6 possibilities. 1 divided by 6 is one-sixth, which is equivalent to 0.167 or 16.7%. So that would be right here on our probability line. That means it isn't likely that I would roll a 3 for instance, but it's just as likely as rolling any other number. And since all 6 numbers have the same probability, each number should come up about as often as the others. To see if they do, I'm going to conduct some trials. That's an excellent argument. Allow me to deliberate… GUILTY! Actually, when dealing with probability, a trial, which can also be called an experiment, is a process that has a random outcome, like tossing a coin, or rolling dice, or spinning a spinner. And the outcome of a trial is what happens in that particular trial, like flipping heads, or rolling a 3. So I'm going to conduct several trials by rolling a die multiple times and keeping track of how many times I roll each number. Aha! You said that each number was going to come up just as often as the other numbers. But look! There's more 2's than there are 5's. How do you explain that? Well remember, we're dealing with things that are random. They're unpredictable. We can't know exactly what will happen, just what will happen on average. So now I have to calculate the average? Well, when we say on average, we mean that the more trials you do, the closer you get to the expected probabilities. Keep watching… There! Now that we've done a lot of trials, you can see that our totals are much closer to what you'd expect them to be. I guess you're probably right! That's one of the really important things to keep in mind about probability. If you do just a few trials, the results might not end up very close to what you'd expect. In fact, they could be way off. But if you do more trials, you increase your chances of reaching the expected probabilities. There's another thing I should point out. Remember the probability of flipping heads is one-half, and the probability of flipping tails is one-half. The probability of rolling a 1 is one-sixth, and the probability of rolling any other number on a die is one-sixth. If you add up the probabilities for the coin flip, you get 2 over 2, or 1. And if you add up the probabilities for rolling a die, you get 6 over 6, which is also 1. And that's not just a coincidence. If you add up the probabilities of all possible outcomes of a trial, the total is going to be 1, or 100%. Because it is certain that at least one of those possibilities will happen. Let's look at some more examples. For these examples, we'll use a spinner. If we had a spinner with just 6 equally sized sectors, the probabilities would be exactly the same as with dice. So, we want a few more sectors. There, that's more like it. Now we have 16 equally sized sectors. So, what's the probability of spinning a 12? Well, just like with dice, where we had to split up the 100% between all 6 possibilities, we'll do the same thing now, but we'll split it up between 16 possibilities. So, the probability of spinning a 12 is 1 over 16, or about 6%, which is right here on the probability line. We can see that the probability of spinning a 12 is less likely than the probability of rolling a 3. And that makes sense because there are more possible outcomes with our spinner. But, what if we color some of the sectors a different color, and we want to know the probability of spinning a certain color? Now we have 5 sectors colored blue and 11 sectors colored yellow. So, what's the probability of spinning a blue? Remember how with the coin toss, we ended up with the fraction 1 over 2, and with the die roll, we got the fraction 1 over 6? In both cases, we had 1 as the numerator. And that's because we were interested in only one of the possible outcomes, like the probability of flipping heads, or the probability of the number 3 being rolled. But in this case, the top number of our fraction will be 5 because any of these 5 sectors will give us the color we want. And the bottom number will still be the total number of possibilities, which is 16, because that's how many total sectors we have. So the probability of spinning a blue is 5 over 16, or about 31%. That's still considered unlikely, but it's more likely than spinning a specific number. And this method will work for figuring out the probability of any event. You just make a fraction with the numerator as the number of outcomes that satisfy your requirement, and the denominator as the total number of possible outcomes. Let's try the same method to find the probability of spinning a yellow. Our top number should be 11 because there's 11 yellow sectors, and our bottom number should still be 16, so the probability of spinning a yellow is 11 over 16, or about 69%. Now we finally have a probability that's considered likely. And it makes sense because you can see by looking at our spinner that it's more likely to spin a yellow than a blue. And you'll notice, if we add up 5 over 16 and 11 over 16, we get 16 over 16, or a probability of 1. So that's a good sign that we did it right. Let's look at another example. Suppose we have a bag of marbles. There are 3 green marbles, 7 yellow marbles and 1 white marble. If we mix them all up and pull out a marble at random, what's the probability of it being green? Well, the top number of our probability fraction will be 3 because there's 3 green marbles, so there's 3 outcomes that get us what we want. And the bottom number will be 11 because there's a total of 11 possible marbles that we could pull out. So the probability of pulling out a green marble is 3 over 11, or 0.27, or 27%. It's right here on the probability line. That means it's unlikely. And that makes sense because you can see that it would be less likely to pull out a green marble than one of the other ones. Let's try this again for calculating the probability of pulling out a yellow marble. This time, the numerator of our fraction will be 7 because there's 7 yellow marbles. The denominator will still be 11 because there's still 11 marbles total. So the probability of pulling out a yellow marble is 7 over 11, or 0.64, or 64%. Another example of an event that's likely. How about pulling out the white marble? Well, the top number will be 1 since there's only 1 white marble, and the bottom number is still 11. So the probability of pulling out a white marble is 1 over 11, or 0.09, or 9%. Not very likely. And if we add up these probabilities, we get 11 over 11, or 100%, just as we expected. Alright, so you should have a pretty good handle on basic probability now. You just have to remember to make a fraction with the numerator being the number of outcomes that give you what you want, and the denominator being the total number of possibilities. And we learned about the probability line and that a probability can't be less than 0 or greater than 1, or 100%. We also learned that the more trials or experiments you conduct, the closer your results will get to the expected probabilities. Of course, the way to get good at it is to practice, so be sure to do a lot of problems on your own. As always, thanks for watching Math Antics and I'll see ya next time. And I sentence you to… 5 years hard labor! Learn more at www.mathantics.com