Complementary Probability

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Can You Roll A 6?

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QUESTIONS — PART 1

Once upon a time, a wealthy Frenchman known as the Chevalier de Méré was fond of betting that, if four dice were rolled, the number 6 would turn up at least once. He made so much money that his gambling buddies refused to play with him anymore. So he invented another dice game where he started losing all the money he won before, but his gambling buddies liked it much better. In 1654, he asked his friend Pascal, famous for his triangle, why one dice game was profitable and the other was not. Pascal liked writing letters to his other friend Fermat, a lawyer who liked to prove things about number sequences. From those letters, the equivalent of text messages these days, was born the mathematical theory of probabilities which soon became the science of statistics with which all rational governments and corporations predict the future. Today, you can thank a pair of dice cubes for your outrageous health insurance.

At first, Pascal and Fermat tried to solve the four dice problem the direct way by adding the probabilities of rolling a 6 on each die: 1/6 + 1/6 + 1/6 + 1/6 = 4/6 = 2/3. But is this right? Experimenting with 4 dice will show this is incorrect. (Roll the mouse over the 4 dice twelve times. How many times does a 6 appear?) If 2/3 were correct, then the probability of rolling a 6 with seven dice would be 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 7/6. But probabilities cannot be greater than 100%. It's possible to roll a die 100 times and not roll a 6 — not likely, but possible. And so, Pascal and Fermat abandoned the direct approach.

Then they realized that it would be faster and easier to determine the chance that none of the four dice would turn up with a 6. They could then simply subtract this probability from 1, and voilà, they would have their answer. Why? Because the probability that something WILL happen plus the probability it will NOT happen is always exactly 1. In symbols, this could be written as

P(x)+P(~x) = 1.

And thus was born the idea of COMPLEMENTARY PROBABILITY. This idea enabled them to answer the following questions. Can you answer them too?
1. If a die is rolled once, what is the probability that it will NOT turn up 6?
2. What is the probability that, if a die is rolled four times, it will not turn up 6 on any of the rolls?
3. What is the probability that, if a die is rolled four times, the number 6 will turn up at least once?
4. Would it be better to bet for or against this happening?
Having solved de Méré's first dice problem, they turned to the second one: if two dice are rolled 24 times, what is the probability a sum of 12 will turn up at least once? These are the questions they asked to solve the second one:
1. If a pair of dice is rolled once, what is the probability that it will not turn up 12?
2. What is the probability that if two dice are rolled 24 times, they will not turn up 12 on any of the rolls?
3. What is the probability that if two dice are rolled 24 times, a sum of 12 will turn up at least once?
4. Would it be better to bet for or against this happening?
5. Did de Méré win a lot of money with this bet?

Have you ever wanted to ask someone an embarassing question, but knew they would not likely answer truthfully? What if the question you would like to ask is "Did you vote for Obama?" You may not want to ask that question to folks in small Tennessee towns. But you could ask them to flip a coin before answering. If they voted for Obama and the coin lands heads, the person is told to answer "yes". Otherwise, they are to answer "no". This way they could hide their true answer half the time. But if you know a little probability, you can use their answer anyway.

Suppose the actual probability someone voted for Obama in Nashville is 54%.
What is the probability that someone answering the question will answer "yes"?
What is the probability that someone answering the question will answer "no"?
Now suppose that a survey is taken throughout Tennessee.
The surveyed flipped a coin as described above,
and 22% of the answers to the question "Did you vote for Obama?" are "yes".
What would you conclude that the percentage of people who actually voted for Obama in Tennessee would be?
Suppose the actual probability someone voted for Obama in the U.S. is 52%.
What is the probability that someone answering the question will answer "yes"?
What is the probability that someone answering the question will answer "no"?
Now randomly select 100 students at TSU, and ask them if they voted for Obama.
The students flip a coin as described above, and 45% of them answer "yes".
What would you conclude that the percentage of students who voted for Obama at TSU would be?
From this percentage, estimate the percentage that voted for McCain.
If you estimate that 2% of the students voted for an independent candidate like Ralph Nader (Yeah, Right), then what percentage voted for McCain?

Complementary Probability

LET'S MAKE A DEAL

Back in the old days of Let's Make A Deal, Monty Hall would show a contestant three doors. Behind two of the doors were farm animals of varying species. Behind the other door was a little red sports car with a blond of your gender preference. The contestant would guess which door veiled the prize, and Monty would open another door revealing a donkey, a goat, a sheep, or a whatever. Then Monty would ask the contestant if he or she wanted to stick or switch doors. Depending on whether the contestant stuck to the original choice, or switched to the other unopened door, he or she went home with style or sty. They just don't make TV like they use to.

DIRECTIONS

Click a door to pick a door.
Play the Monty Hall simulation above 100 times.
STICK to your original guess for all 100 times.
When you have played the 100th time, screencopy and save your results.
CAREFUL when you get to 100. The counter resets after playing 100 times.
Play the Monty Hall simulation again 100 times.
This time SWITCH each time from your original guess after Monty opens a door.
When you have played the 100th time, screencopy and again save your results.
Email both the STICK and the SWITCH copies to your instructor.
Do not use anyone else's copy,
Or you and the one you copied from will receive no points.
Now answer the questions in the table below for 10 points maximum.
After you have completed the table, screencopy it and email it to your instructor.
Due by noon one week from today.

QUESTIONS

What is the probability that you will pick the car door on your first guess?
What is the probability that you will not pick the car door on your first guess?
After Monty opens a door, what is the probability that your door conceals the sports car?
After Monty opens a door, what is the probability that your door does not conceal the sports car?
In the simulation, how many times did you win when you Stuck to your original guess?
In the simulation, how many times did you win when you Switched from your original guess?
Do these last two answers agree with your two answers to #3 and #4?
Would you like to stick with your answers to #3 and #4, or switch?
Suppose instead of three doors, there were ten doors. Suppose you guess Door #10, and Monty opens eight of the remaining doors. What is the probability that you picked the car door on your first guess?
Should you stick to Door #10 or switch?

Note that answers with numbers require 3 decimal precision.

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Last Modified 11/13/08 5:17 PM