Probability Games

Here are some probability games that you can try out during the latter 1.5 hours of class.

The purpose of these games are as follows:

1. Refresh your memory of probability rules.
2. Get your mind thinking about probability problems, which may come handy in later classes.
3. Give you a warm-up to Bayesian thinking.

The problems are directly taken from the Think Bayes and Allen Downey's website, so while theoretically you can always go and look at the answer, you'll benefit most from working it out with others on the blackboard, or if you're feeling up for a challenge, by coding the solution up.

Problem 1: The cookie jar problem

Suppose there are two bowls of cookies. Bowl 1 contains 30 vanilla cookies and 10 chocolate cookies. Bowl 2 contains 20 of each.

Now suppose you choose one of the bowls at random and, without looking, select a cookie at random. The cookie is vanilla. What is the probability that it came from Bowl 1?

Problem 2: The M&M problem

M&M’s are small candy-coated chocolates that come in a variety of colors. Mars, Inc., which makes M&M’s, changes the mixture of colors from time to time. In 1995, they introduced blue M&M’s. Before then, the color mix in a bag of plain M&M’s was 30% Brown, 20% Yellow, 20% Red, 10% Green, 10% Orange, 10% Tan. Afterward it was 24% Blue , 20% Green, 16% Orange, 14% Yellow, 13% Red, 13% Brown.

Suppose a friend of mine has two bags of M&M’s, and he tells me that one is from 1994 and one from 1996. He won’t tell me which is which, but he gives me one M&M from each bag. One is yellow and one is green. What is the probability that the yellow one came from the 1994 bag?

Problem 3: The Monty Hall problem

Monty Hall was the original host of the game show Let’s Make a Deal. The Monty Hall problem is based on one of the regular games on the show. If you are on the show, here’s what happens:

• Monty shows you three closed doors and tells you that there is a prize behind each door: one prize is a car, the other two are less valuable prizes like peanut butter and fake finger nails. The prizes are arranged at random.
• The object of the game is to guess which door has the car. If you guess right, you get to keep the car.
• You pick a door, which we will call Door A. We’ll call the other doors B and C.
• Before opening the door you chose, Monty increases the suspense by opening either Door B or C, whichever does not have the car. (If the car is actually behind Door A, Monty can safely open B or C, so he chooses one at random.)
• Then Monty offers you the option to stick with your original choice or switch to the one remaining unopened door.

The question is, should you “stick” or “switch” or does it make no difference?

Problem 4: The Elvis Presley problem

Elvis Presley had a twin brother who died at birth. What is the probability that Elvis was an identical twin? To answer this one, you need some background information: According to the Wikipedia article on twins:

"Twins are estimated to be approximately 1.9% of the world population, with monozygotic twins making up 0.2% of the total---and 8% of all twins."