Multiplication rule for independent events
On a multiple choice test, problem 1 has 4 choices, and problem 2 has 3 choices. That should be choices. Each problem has only one correct answer. What is the probability of randomly guessing the correct answer on both problems? Now, the probability of guessing the correct answer on each problem-- these are independent events. So let's write this down. The probability of correct on problem number 1 is independent. Or let me write it this way. Probability of correct on number 1 and probability of correct on number 2, on problem 2, are independent. Which means that the outcome of one of the events, of guessing on the first problem, isn't going to affect the probability of guessing correctly on the second problem. Independent events. So the probability of guessing on both of them-- so that means that the probability of being correct-- on guessing correct on 1 and number 2 is going to be equal to the product of these probabilities. And we're going to see why that is visually in a second. But it's going to be the probability of correct on number 1 times the probability of being correct on number 2. Now, what are each of these probabilities? On number 1, there are 4 choices, there are 4 possible outcomes, and only one of them is going to be correct. Each one only has one correct answer. So the probability of being correct on problem 1 is 1/4. And then the probability of being correct on problem number 2-- problem number 2 has three choices, so there's three possible outcomes. And there's only one correct one, so only one of them are correct. So probability of correct on number 2 is 1/3. Probability of guessing correct on number 1 is 1/4. The probability of doing on both of them is going to be its product. So it's going to be equal to 1/4 times 1/3 is 1/12. Now, to see kind of visually why this make sense, let's draw a little chart here. And we did a similar thing for when we thought about rolling two separate dice. So let's think about problem number 1. Problem number 1 has 4 choices, only one of which is correct. So let's write-- so it has 4 choices. So it has 1-- let's write incorrect choice 1, incorrect choice 2, incorrect choice 3, and then it has the correct choice over there. So those are the 4 choices. They're not going to necessarily be in that order on the exam, but we can just list them in this order. Now problem number 2 has 3 choices, only one of which is correct. So problem number 2 has incorrect choice 1, incorrect choice 2, and then let's say the third choice is correct. It's not necessarily in that order, but we know it has 2 incorrect and 1 correct choices. Now, what are all of the different possible outcomes? We can draw a little bit of a grid here. All of these possible outcomes. Let's draw all of the outcomes. Each of these cells or each of these boxes in a grid are a possible outcome. You could-- you're just guessing. You're randomly choosing one of these 4, you're randomly choosing one of these 4. So you might get incorrect choice 1 and incorrect choice 1-- incorrect choice in problem number 1 and then incorrect choice in problem number 2. That would be that cell right there. Maybe you get this-- maybe you get problem number 1 correct, but you get incorrect choice number 2 in problem number 2. So these would represent all of the possible outcomes when you guess on each problem. And which of these outcomes represent getting correct on both? Well, getting correct on both is only this one, correct on choice 1 and correct on choice-- on problem number 2. And so that's one of the possible outcomes and how many total outcomes are there? There's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, out of 12 possible outcomes. Or since these are independent events, you can multiply. You see that they're 12 outcomes because there's 12 possible outcomes. So there's 4 possible outcomes for problem number 1, times the 3 possible outcomes for problem number 2, and that's also where you get a 12.