Independent events example: test taking

on a multiple-choice test problem one has four choices and problem two has three 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 correct on on problem on number one on problem number one is independent or let me write it this way probability of correct on number one and probability and probability of correct on number two on problem two are independent are independent are independent which means that the outcome of one of the events of guessing on the first on the first problem isn't going to affect the probability of guessing correctly on the second problem independent independent events so the combined the probability of guessing on both of them so that means that the probability that the probability of being correct on guessing correct on one and number two is going to be equal to the product of these probabilities and we're going to see why that is visually in a second which is going to be the probability of correct on number one times the probability the probability of being correct on number two now what are the what are each of these probabilities on number one there are four choices there are four 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 one is one-fourth and then the probability of being correct on problem number two not problem number two 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 two is one-third probability of guessing correcto number one is 1/4 the probability of doing on both of them on both of them is going to be its product so it's going to be equal to 1/4 times 1/3 is is 1/12 now to see kind of visually why this makes sense let's draw a little chart here and we did a similar thing for when we thought about when we thought about rolling two separate dice so let's think about problem number one problem number one has four choices only one of which is correct so let's write so it has four choices so it has one let's write incorrect choice one incorrect choice two incorrect choice three and then it has the correct choice over there so those are the four 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 two problem number two has three choices only one of which is correct so problem number two has incorrect choice one incorrect choice two and then let's say the third choice is corrected that's not necessary in that order but we know it has to incorrect and one correct choices now what are all of the different possible outcomes we can draw a little bit of a grid here you can draw 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 our possible outcome you could you're just guessing your randomly choosing one of these for your randomly choosing one of these four so you might get incorrect choice one and incorrect choice one in in incorrect choice in problem number one and then incorrect choice in problem number two that would be that cell right there maybe you get this maybe you get problem number one correct but you get incorrect choice number two and problem number two 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 one and correct on choice on problem number two and so that's one pot 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 these are independent events you can multiply you see that there are 12 outcomes because there's 12 possible outcomes so there's four possible outcomes for problem number one times the three possible outcomes for problem number two and that's also where you get a 12