 When dealing with any complicated probability problem, which is all of them, it's helpful to organize information into a contingency table. A two-way contingency table organizes data into rows and columns, where the rows represent mutually exclusive outcomes, and the columns also represent mutually exclusive outcomes. And finally, all possible outcomes are included in the table. For example, suppose a group of 150 students has interviewed to determine their taste preferences between chocolate chip cookies and kale. There are 50 math majors and 100 English majors. No one is a double major. 37 other students prefer chocolate chip cookies, while 81 English majors prefer kale. Find the probability a randomly selected student is a math major who likes kale. Now, we're told that no one is a double major. And so it appears that English major and math major are, in this experiment, mutually exclusive, so we can make those the column entries. Likewise, prefers chocolate and prefers kale are also mutually exclusive. You can't prefer both, so we can make those the row entries. It's helpful to extend the table to include the totals as well. Now, when constructing a contingency table, the most important thing to remember is the books got balanced. In other words, all of our totals have to be consistent with the given information. So the total of all students is 150. Those are the English and math majors and the chocolate and kale preferers. We know there are 50 math majors and 100 English majors in total, so we can set these numbers in the totals column. We also know that a total of 37 students prefer chocolate chips, so we can set that in the totals row of chocolate. Finally, we know that 81 of the English majors prefer kale, which we put in the English majors row, kale column. Now, since there are 150 students altogether and 37 of them prefer chocolate chip, then the books got a balance. That means that 150 minus 37 or 113 prefer kale. Since 81 English majors prefer kale and there are 100 altogether, then 100 minus 81 or 19 prefer chocolate chip. Now, we notice that there are 37 students who prefer chocolate chip. We've identified 19 of them are English majors and so the rest must be math majors. Since there are 50 math majors and 18 of them prefer chocolate chip, then 50 minus 18 equals 32 should prefer kale. And now that our contingency table is completely filled out, we can calculate any probability. So let's consider of the 150 students, there are 32 math majors who prefer kale, so the probability a randomly selected student is a math major who prefers kale is 32 150th. We can do the same thing if we have percentages. For example, suppose 37% of a company's customers own SUVs and 47% own hybrids. Could 10% of the customers own neither? To construct the contingency table, note that owns SUV and owns hybrid are not mutually exclusive, so they shouldn't be in the same row or column. Now, since we want all possibilities to show up, then owns SUV and doesn't own SUV are complimentary, so no SUV should be the second column label. And similarly, owns hybrid and doesn't own hybrid are complimentary, so no hybrid should be the second row label. And again, it's convenient to extend to a total's column and row. Now, we're given percentages and since we're given percentages, it's convenient to suppose we have 100 customers. Then 37% of 100 or 37 own SUVs, 47% of 100 or 47 own hybrids, and let's see if it's possible to complete the table if 10% of 100 or 10 own neither. Since there are 100 customers and 37 of them own SUVs, then the number who don't own an SUV will be. Likewise, since there are 100 customers and 47 own a hybrid, then the number who don't own a hybrid will be. Since there are 63 who don't own an SUV and 10 who own neither a hybrid nor SUV, then the number who do own a hybrid will be. And we have a problem because the number who own a hybrid but no SUV is greater than the number who own a hybrid. And so we can conclude that it's not possible to have 10% of the customers to own neither hybrids nor SUVs. So suppose we go and find the correct data and let's say that we know that 5% of a company's employees own both a hybrid and an SUV while 8% own neither. If 77% own an SUV, what percentage own a hybrid? And again, since we're given whole number percentages, let's assume we're working with 100 employees so we can work with actual numbers. So again, we'll set up a contingency table. Now we're told that 5% of 100 or 5 own a SUV and a hybrid, so they go in this column, 8% of 100 or 8 own neither, so they go in this column, and 77% of 100 or 77 own a SUV, so they go in this column in the totals row. So now we can start filling out the table. Since 77 own a SUV, then the number who don't own a SUV is since 23 do not own a SUV and 8 own neither an SUV nor a hybrid, then the number who own a hybrid is and since 5 own both a hybrid and an SUV and 77 own an SUV, then the number who own an SUV but not a hybrid will be. And finally we can round out the totals. We note that 5 plus 15 or 20 own a hybrid, 72 plus 8 or 80 don't own a hybrid. It's also useful to verify the cross totals. We computed the numbers who own or don't own a hybrid, 20 and 80. This should add to the total number, so we check 20 plus 80 equals 100. So now let's find the percentage who own a hybrid. So we see that of our employees, 20 out of 100 or 20% own a hybrid.