Basic theoretical probability
Find the probability of pulling a yellow marble from a bag with 3 yellow, 2 red, 2 green, and 1 blue-- I'm assuming-- marbles. So they say the probability-- I'll just say p for probability. The probability of picking a yellow marble. And so this is sometimes the event in question, right over here, is picking the yellow marble. I'll even write down the word "picking." And when you say probability, it's really just a way of measuring the likelihood that something is going to happen. And the way we're going to think about it is how many of the outcomes from this trial, from this picking a marble out of a bag, how many meet our constraints, satisfy this event? And how many possible outcomes are there? So let me write the possible outcomes right over here, so possible outcomes. And you'll see it's actually a very straightforward idea. But I'll just make sure that we understand all the words that people might say, so the set of all the possible outcomes. Well, there's three yellow marbles. So I could pick that yellow marble, that yellow marble, or that yellow marble, that yellow marble. These are clearly all yellow. There's two red marbles in the bag. So I could pick that red marble or that red marble. There's two green marbles in the bag. So I could pick that green marble or that green marble. And then there's one blue marble in the bag. There's one blue marble. So this is all the possible outcomes. And sometimes this is referred to as the sample space, the set of all the possible outcomes. Fancy word for just a simple idea, that the sample space, when I pick something out of the bag, and that picking out of the bag is called a trial, there's 8 possible things I can do. So when I think about the probability of picking a yellow marble, I want to think about, well, what are all of the possibilities? Well, there's 8 possibilities, 8 possibilities for my trial. So the number of outcomes, number of possible outcomes, you could view it as the size of the sample space, number of possible outcomes, And it's as simple as saying, look, I have 8 marbles. And then you say, well, how many of those marbles meet my constraint, that satisfy this event here? Well, there's 3 marbles that satisfy my event. There's 3 outcomes that will allow this event to occur, I guess is one way to say it. So there's 3 right over here, so number that satisfy the event or the constraint right over here. So it's very simple ideas. Many times the words make them more complicated than they need to. If I say, what's the probability of picking a yellow marble? Well, how many different types of marbles can I pick? Well, there's 8 different marbles I could pick. And then how many of them are yellow? Well, there's 3 of them that are actually yellow.