Wednesday, June 24, 2009

Please Help Dorothy Go Home - A Probability Fantasy for Middle School and Beyond

Summer vacation is an appropriate time for fantasy. Enjoy the hiatus!

The following investigation is not intended to be a math contest challenge. It reviews fundamental principles of probability and you might want to bookmark it for the fall. We can also simulate the first problem using the programming capabilities of a graphing calculator. I may post a simple program for this later on.

The wizard will let Dorothy go home if she can pass three challenges.

He shows Dorothy 3 playing cards, 2 of which are black and one is red. He shuffles them and turns them face down. "Dorothy, here's your first challenge."

"You will pick a card. If it's red the game ends, you win the game. If it's black, I will remove the card and you will pick a card from the remaining two. If it's red you still win! Ah, but if it's black again you and Toto and your weird friends will remain here for at least one more month."

Well, Dorothy won the game and said, "Now, I want to go home!" But the crafty wizard said, "You weren't listening carefully, Dorothy. I never said you can go home if you won the game. You've only passed the first challenge. You must still pass two more." "That's not fair!" Dorothy protested but the wizard makes his own rules in Oz.

"Alright, Dorothy, you won the game but you knew the odds were in your favor since you had two chances to win. Here's your next challenge:

"What was the probability of your winning and you must give me two correct but different methods?"

Dorothy asked, "These are the remaining challenges, so if I get them right, I can go home, yes??"
"I will not lie to you, Dorothy. This is your 2nd challenge. There will still be one more."

Dorothy was upset but knew she had no choice but to trust him. She thought about the problem for a minute and replied, "The probability of my winning was 2/3. I know I'm right!"
"Very good, Dorothy, but you must explain that answer two different ways." Fortunately, Dorothy was a very responsible middle school student back in Kansas and had learned the methods of compound probabilities and the idea of complementary events (this is a fantasy after all!).

Dorothy was able to provide two correct methods. Can you?

"Very good, Dorothy! You only have one more challenge to conquer and you can go home.
This time there are N cards, one of which is red while the remaining cards are black. N is a positive integer greater than 1. Same rules as before. The cards are shuffled and laid out face down. You pick a card. If it's red the game is over and you win. If it's black, the card is removed and you try again. The game continues until you pick the red card. The only way to lose the game is if you pick all the black cards and the last card remaining is red."

"In terms of N, what is the probability that you will win? Oh, yes, you again have to show two different methods in detail on this magic board over here."

This time, Dorothy needs your help. She can guess the formula but she needs our help to show two ways to derive it. Please help Dorothy go home!


mathmom said...

Nice, but beyond most middle-schoolers I believe (at least the general case).

Does computing the probability of winning, and computing the probability of losing and subtracting from 1 count as 2 separate methods?

For the simple case students might try to just enumerate the possible outcomes, but it's a bit of a leap to prove that they're all equally likely.

Dave Marain said...

I would count those as 2 different methods.
Yes, the general case may be beyond them but working with a few special cases like 3, 4 and 5 might lead to a simple generalization. The general formula for losing is 1/n and the simplicity of that result suggests there might be a clever approach.

mathmom said...

I think some of them could figure out the general formula, but most could probably not justify it.

jd2718 said...

I left a comment, but I think blogger ate it.

Let's modify things a bit. Have the kid keep drawing cards, even after they win. Won't change the probability, they've already won. But it simplifies the game. Now each play consists of drawing all but one card.

The modified game is equivalent to the original game. This modification may be difficult to explain, but now the math is accessible to much younger children.