When to Add, When to Multiply?
Date: 10/30/2001 at 16:23:25 From: Aliza Subject: Probability I am studying probability at high school. When do you add and when multiply to compute probability?
Date: 10/31/2001 at 06:59:39 From: Doctor Mitteldorf Subject: Re: Probability Dear Aliza, You're looking for rules that will get you through. t's not going to work. In probability more than in any other field, you have to understand the situation, and have a feel for it. Doing lots of problems will help, even if you feel insecure trying to work them at first. It helps especially if you do your best to think about the problems on your own. Try this and that - see what makes sense to you. The calculation is the easy part, and knowing what to add or when to multiply is what it's all about. Remember that probabilities are all numbers less than 1. When you multiply them together, they get smaller. When you multiply several together, they can rapidly get a lot smaller. Now think about unlikely events: If it's unlikely that one such event occurs, it's "doubly" unlikely that two will occur. You multiply probabilities together to compute the unlikeliness of this coincidence - two unlikely events both occurring together. Probabilities are like any other positive number; when you add them together, they get larger. Adding the probabilities for two unlikely events gives you a larger probability that either one or the other will occur. When you're looking for the probability that two events, A and B, will BOTH occur, the probability of this coincidence is small, and you multiply the separate probabilities of A and B to get a smaller number. When you don't care which happens - either A or B - you can add the probabilities to find the separate probability that one or the other will happen. So now I've done it. First I told you that there are no rules for telling you when to multiply and when to add; then I gave you a rule that tells you when to multiply and when to add. But now that I've given you the rule, I should also tell you about the exception. This is the case that really makes the subject hard to think about. Think about whether it's logically possible for both A and B to occur. For example, two people have bought raffle tickets. If A wins the grand prize, then B cannot win, and vice versa. In this case, it makes perfect sense to add the probability of A winning to the probability of B winning, and the result is the probability that "either one of them" will win. If there are 1,000 raffle tickets, then A has a 1/1000 chance of winning and B has a 1/1000 chance of winning. Together, their chance of winning is exactly 2/1000. On the other hand, think about a situation where A or B or BOTH are all possibilities. What is the probability that A or B has a birthday on Halloween? Let's assume the probability that A has a birthday on Halloween is 1/365, and that B has a birthday on Halloween is 1/365. Your first thought is that the probability of either A or B having a birthday on Halloween is 2/365. This seems right until you ask what the probability is for 3 people. Is it 3/365? For 100 people is it 100/365? For 1000 people, what is the probability that one or more of them has a birthday on Halloween? Is it 1000/365? No - that doesn't make sense, because 1000/365 is a number greater than one. The problem with adding 1/365 + 1/365 = 2/365 is that we've DOUBLE COUNTED the probability that both A and B have their birthdays on Halloween. We can correct the answer by subtracting that tiny coincidental probability: the right answer is 1/365 + 1/365 - 1/(365*365). How would you go on to find the probability that either A or B or C has a birthday on Halloween? Now what have you double-counted? This is harder to think about, because you've double-counted (A and B), but you've also double-counted (B and C) and (A and C). And what about the REALLY unlikely possibility that A and B and C all have birthdays on Halloween? Have you double-double counted it? Or have you double counted the double counting, in which case a minus times a minus is a plus, and you need to add it back? This is confusing, I know, and there are only 3 people. We need a better way to think about it, especially if we're going to be able to handle the question about 1000 people. Here's a trick that will solve the problem more easily - but the drawback is that it takes us back to the situation where adding and multiplying can become confused, and you can be left wondering whether to add or to multiply. Think about the probability that A does NOT have a birthday on Halloween. That's 364/365. Same for B. If you want the probability of the "coincidence" that they both don't have birthdays on Halloween, you can get that probability by multiplying 364/365 * 364/365. This is exactly the right answer. The situation that either one or the other or both has a birthday on Halloween is exactly the opposite of this, so you can subtract from one 1 - 364/365 * 364/365 and get exactly the right answer. So here it is: the situation where your first thought was that adding 1/365 and 1/365 should work, but actually you needed to multiply, not to add. And to make matters more confusing, the difference between the two answers is really tiny, because 1 - 364/365 * 364/365 is almost the same as 2/365. I'll finish with a puzzle for you. I've given you two ways to get the right answer: the other one was 2/365 - 1/(365*365). Which is the right right answer? - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/
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