 Suppose you have two events a and b. There are three possibilities. They are mutually exclusive, only one of them can occur at a time. They're complementary, exactly one of them must occur. The third possibility is that either, both, or neither could occur. The event both a and b is called a conjunction. That's when they happen at the same time. Meanwhile, the event either a or b is called a disjunction. That's when we care if either of the two events occur. So we can develop a formula that relates the disjunctions and conjunctions, but remember, don't memorize formulas. Understand concepts. And here the useful concept is the contingency table. A contingency table is actually most useful when the rows and columns correspond to complementary events, a or not a, b, or not b. And if we imagine that these events occur with frequencies p, q, m, and n, we can extend our totals. And now let's consider probabilities. We note that the number of times a and b occur, that's p times. Well, that's really the number of times a occurs, plus the number of times b occurs. But if we just add these two, we end up with a bunch of extra terms. So we have to subtract those off and we see that these numbers correspond to the times when either a or b occurs. And since this is out of a total of p, plus q, plus m, plus n, we can get the empirical probability by dividing all terms by p, plus q, plus m, plus n, giving us the relationship. And so the probability of a and b occurring is the probability a occurs, plus the probability b occurs, minus the probability that either one of them occurs. Now one useful thing to remember is that if you get a new tool, make sure it works on the old problems. So for example, suppose a and b are mutually exclusive, then the probability of both a and b is zero and our formula will give. We can rearrange this, which is what it's supposed to be. Likewise, if a and b are complementary, then again the probability they both occur is going to be zero, so we find. But since they're complementary events, one of them must occur, giving us the probability of either a or b equal to one. And so we find, which is what we found earlier. So for example, suppose we know that we have 120 patients admitted to a hospital, 73 with broken limbs, 53 with burns, and 23 with both. Let's find the probability a randomly selected patient had either a burn or a broken limb. Now we have this nice new formula, but let's find the probability by constructing a contingency table. So our possibilities, either somebody has a broken limb, or they don't, or they have a burn. Or they don't. As we know, there's 120 patients admitted to the hospital. We know how many had broken limbs, so we know that total. We know how many had burns, so we know that total. And we know the number of patients who had both burns and broken limbs. So we can fill out the rest of the table. And the patients who had either a burn or a broken limb are this set. And so we see that 23 plus 30 plus 50, 103, had either a broken limb or a burn. And so we can find our probability. Of course, we do have a formula. So our events are broken limbs or burns. So the probability of a broken limb and a burn is the probability of a broken limb plus the probability of a burn minus the probability of one or the other. We can find our probabilities and solve for the one we don't know. As this example suggests, this formula is not actually very useful if you're willing to construct the contingency table. In fact, let's consider this problem. An executive at Megacor Insurance Company determines the probability a policyholder owns an SUV but not a hybrid is 0.3, while the probability they own a hybrid but not an SUV is 0.35. Finally, they notice that 55% of their policyholders don't own hybrids. They ask you to find the probability a policyholder owns both an SUV and a hybrid. So you have this handy formula for finding the probability of a conjunction. But if you only have a hammer, you must treat every problem like a nail. So we could use our formula and what we're given are the following probabilities and the thing to notice is that these are none of the probabilities we need. And that means using our formula is going to require that we go out and find a lot more probabilities. So let's construct a contingency table instead. Since we're given probabilities to two decimal places at a percentage, let's consider a set of 100 customers. So suppose we have 100 customers. Then we can break these customers down into whether they own an SUV or not, and whether they own a hybrid or not, and we can extend our totals. Since the probability they own an SUV but not a hybrid is 0.3, then we'd expect that 0.3 of 100 or 30 own SUVs but not hybrids. Since the probability they own a hybrid but not an SUV is 0.35, then we'd expect that 0.35 of 100 or 35 own hybrids but not SUVs. And finally, since 55% of policy holders don't own hybrids, then 55 don't own hybrids. We can complete the rest of the table. Since 55 don't own hybrids, the number who do own hybrids will be Since 55 don't own hybrids and 30 of them own SUVs, the number who own neither hybrids nor SUVs will be Since 45 own hybrids and 35 don't own SUVs, the number who do own SUVs is Extending our totals gives us And now that we have the contingency table, we can find our empirical probabilities. So 10 of the 100 own both SUVs and hybrids, giving us the empirical probability 10-100s. Or we could use our formula. So there's two events, someone owns an SUV or they own a hybrid. So the probability of both is the probability of the one, plus the probability of the other, minus the probability of either. And we can read those probabilities off our table. And here's the thing, once you've produced the table, there's no compelling reason to use our formula.