Fruit Juice Tasting

^{Math Puzzles of the Month}
Fruit Juice Tasting

In this puzzle, we challenge you to find the probability of guessing what flavor of juice is in each of several cans without looking at the labels.

You have been invited to a fruit-juice-tasting party. There are several cans of juice, each of a different flavor. The cans are covered with paper so that their labels are hidden.

You have a list of all the different flavors. You have to guess what flavor is in each can without looking at its label. In other words, you have to match each flavor on the list with a can. You can't guess a flavor more than once.

Our math puzzle this month is in two parts.

Part 1

To start you have two cans of juice. What is the probability that you will guess correctly? (Since this is a math puzzle, let's assume that you really can't taste the difference so you are just making random guesses.)

With 3 cans, what's the probability of guessing all of them correctly? What if there are 4 cans of juice? How about 10? How about 100?

Can you come up with a general formula for the probability of guessing all the flavors correctly with N cans of juice?

Part 2

Now let's look at the probability of getting all the guesses wrong. If there are only two cans of juice, then you guess either both correctly or both incorrectly. But with three cans, you could get one right and the other two wrong. What is the probability of getting all three wrong?

With 10 cans of juice, what is the probability that you will get none of them right? How about with 100 cans?

Can you come up with a general formula for the probability of getting no matches at all when trying to guess the flavors of N cans of juice?

Finally, what is the probability of getting just one right?

After you've tried this for yourself, check our solution.