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Applications of Probability

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Probability to Analyze Fairness and Decisions

Your friend Jeff is on a game show hoping to win a car. The host of the game show reveals three doors, and tells Jeff that the car is behind one of them. Behind the other two doors are goats. The game show host knows where the car is. 

The game show host tells Jeff to pick a door and he does. Then, the game show host opens one of the doors that Jeff did NOT choose to reveal a goat. 

The game show host asks Jeff is he wants to switch doors. Jeff says yes, because he believes that his chance of winning is greater if he switches his choice. 

Did Jeff make the right decision?

Watch This

This video helps to demonstrate the Monty Hall Problem from the Concept.

http://www.youtube.com/watch?v=mhlc7peGlGg

Guidance

The word “fair” is frequently used informally, often having to do with the inequities that exist in our world:

  • “It's not fair that Mark has more video games than me.”
  • “It's not fair that some people don't have enough money for food.”
  • “It's not fair that I have to take a Spanish class.”

Fairness in the above situations can be a matter of opinion. 

The word “fair” is also used formally, in the context of games and making choices. A basic game of chance is considered fair if every player has an equal probability of winning. A choice is fair if all possible options have an equal probability of being chosen. You can use your knowledge of probability to analyze the fairness of games and make fair choices.

In complex games or situations like the Concept problem, players have choices to make and will often have strategies for increasing their chance of winning. You can use your knowledge of probability to analyze different strategies for a given game.

Example A

A bag contains 10 red marbles and 1 silver marble. You and your brother decide that you will reach into the bag and select a marble. If you choose a red marble then you have to do the dishes and if you choose a silver marble then he has to do the dishes. Is this fair?

Solution: To be fair, both you and your brother should have an equal chance of having to do the dishes. This  means the probability that you will do the dishes should be \frac{1}{2} (50\%) and the probability that your brother will do the dishes should be \frac{1}{2} (50\%) .

  • You will have to do the dishes if you choose a red marble. There are 10 red marbles and 11 marbles total. The probability of you doing the dishes is \frac{10}{11} \approx 91\% .
  • Your brother will have to do the dishes if you choose the silver marble. There is 1 silver marble and 11 marbles total. The probability of your brother doing the dishes is \frac{1}{11} \approx 9\% .

This is definitely not a fair way to decide who will do the dishes, because you are much more likely to end up doing the dishes.

Example B

Your math class has 32 students. Your teacher, Ms. Peters, needs to choose 5 people randomly to make a presentation. How can she use her calculator to help her choose 5 people at random?

Solution: Random in this sense means that Ms. Peters wants each student to have an equal chance of being chosen. If she just chooses 5 people and tries to be random, her choices likely won't actually be random. She might choose people who haven't presented recently, or try to get a mix of boys and girls. 

To make a truly random choice, it helps to use a random number generator. First, she should assign each person in the class a number from 1 to 35 (this could be done in alphabetical order or in any other way). Then, she should use the randInt( function on her graphing calculator. This function produces a given number of random integers between two integers. It is found on a TI-83/TI-84 by pushing the MATH button, then scrolling right to PRB , then scrolling down to 5: randInt( . Back on the home screen, type the lower limit of 1, the upper limit of 35, and 5 (for the 5 numbers that you want produced). Press ENTER to produce 5 random numbers between 1 and 35.

This time, students 34, 32, 6, 19, and 15 should make the presentation. Note that it is possible that the same number could have been chosen twice. If that happened, Ms. Peters could just randomly generate another number. 

Graphing Calculator Randomness: Even the graphing calculator isn't truly random. It is following a formula to create numbers that appear random. Using their default setting, all calculators will produce the same random numbers in the same order, because they are using the same formula. If you wish to change the starting number your calculator is using in its formula to create random numbers, type in a number (any number), then press STO> , then use the  rand function and press ENTER . Now the random numbers your calculator generates will be different from the random numbers generated by another calculator.

Example C

Your history class taught by Mr. Bliss has 35 students. On the first day of school Mr. Bliss asks everyone when their birthday is. You are surprised to learn that two people in your class have the same birthday. You say “I bet if we looked at the birthdays for each history class taught by Mr. Bliss, none of the other classes will have a birthday shared by at least two people.” If Mr. Bliss teaches five history classes, each with 35 students, is this a good bet?

Solution: Consider the probability of a group of 35 people NOT having the same birthday. In order to find this probability you need:

  1. The number of ways to assign different birthdays (from the 365 days in a year) to 35 people.
  2. The number of ways to assign a birthday (from the 365 days in a year) to each of 35 people (repeats okay).

You can use the fundamental counting principle for these calculations

1. To assign a different day to each person, there are 365 choices for person #1, 364 choices for person #2, 363 choices for person #3, and so on.

Number of ways to have 35 distinct birthdays: 365 \cdot 364 \cdot 363 \cdots 333 \cdot 332 \cdot 331

2. To assign a day to each person where repeats are okay, there are 365 choices for person #1, 365 choices for person #2, and so on.

Number of ways to pick birthdays for 35 people:  365 \cdot 365 \cdots 365 \cdot 365=365^{35}

The probability of 35 distinct birthdays can be calculated with the help of a computer to be:

\frac{365 \cdot 364 \cdot 363 \cdots 333 \cdot 332 \cdot 331}{365^{35}} \approx 18.6\%

If the probability of everyone have a different birthday is 18.6%, then the probability of at least two people sharing a birthday (which is a complementary probability), is 100\%-18.6\%=81.4\% . Even though it might seem rare, there is actually a pretty high probability that in a group of 35 people there will be a shared birthday. 

If Mr. Bliss teaches 5 history classes, it's definitely possible that more than one class will have a shared birthday within it. Therefore, you did not make a good bet. 

This is a classic probability problem with a result that surprises most people!

Concept Problem Revisited

As the video demonstrates, Jeff's chance of winning if he switches is \frac{2}{3} . His chance of winning if he doesn't switch is \frac{1}{3} . This can be extremely counter-intuitive. It helps to remember that at the beginning, Jeff has a \frac{1}{3}  chance of choosing the correct door. Since the host knows where the car is and will always open a door with a goat behind it, the other door (the unopened door that Jeff did not originally choose) will have the car behind it \frac{2}{3}  of the time. A good strategy for winning the car in this game is to always switch doors when given the choice. 

Vocabulary

The probability of an event is the chance of the event occurring.

A combination is the number of ways of choosing k  objects from a total of n  objects (order does not matter). The notation for combinations is  nCk or \binom{n}{k} . The formula is nCk=\frac{nPk}{k!}=\frac{n!}{k!(n-k)!} .

A permutation is the number of ways of choosing and arranging k  objects from a total of n  objects (order does matter). The notation for permutations is nPk . The formula is nPk=\frac{n!}{(n-k)!} .

The fundamental counting principle states that for independent events  A and B , if there are n  outcomes in event  A and m  outcomes in event B , then there are n \cdot m  outcomes for events  A and  B together. 

A game or choice is fair , if every person has an equal chance of winning or being chosen.

A choice is random , if every person or object has an equal chance of being chosen.

Guided Practice

1. Use the random number generator to randomly choose 10 numbers between 1 and 100.

2. Ben makes a game involving tossing three coins and noting the sequence of heads and tails. If more heads than tails appear then he wins. If more tails than heads appear then you win. Is this game fair?

3. What if Ben instead tosses four coins? If no heads or an even number of heads appears then he wins. If an odd number of heads appears then you win. Is this game fair?

Answers:

1. On your calculator, use the randInt( function. Enter randInt(1,100,10) and record the 10 numbers that come up. Numbers will vary depending on where you set your calculator to start (as discussed in Example B).

2. To see if the game is fair, calculate Ben's chance of winning and your chance of winning. For the experiment of tossing three coins, the sample space is {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. The probability of Ben winning is \frac{4}{8}=\frac{1}{2}  because 4 of the 8 outcomes involve more heads than tails. The probability of you winning is \frac{4}{8}=\frac{1}{2}  because 4 of the 8 outcomes involve more tails than heads. This is a fair game.

3. The sample space has 16 outcomes: {HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, THHT, THTH, TTHH, HTTH, TTTH, TTHT, THTT, HTTT, TTTT}. The probability of Ben winning is \frac{8}{16}=\frac{1}{2}  because 8 of the 16 outcomes involve 0, 2, or 4 heads. The probability of you winning is \frac{8}{16}=\frac{1}{2}  because 8 of the 16 outcomes involve 1 or 3 heads. This game is fair.

Practice

1. What makes a game fair?

2. Why do people sometimes “flip a coin” to make a decision?

3. You have one prize and 30 people that could win the prize. Describe at least 2 ways to fairly choose who gets the prize.

4. You and your three siblings are trying to decide who gets the last cookie. Describe a way you could use coins to help make a fair decision about who gets the cookie.

5. Use a random number generator to pick 8 numbers at random between 1 and 300.

6. Describe a situation where you might want to use a random number generator.

7. Your friend invents a dice game. You roll two dice. If the sum of the numbers that show up is even, you win. If the sum of the numbers that show up is odd, he wins. Is this game fair? Explain.

8. Your friend changes the dice game. Now, if the sum of the numbers is 6 or less, you win. If the sum of the numbers is 7 or more, he wins. Is this game fair? Explain.

9. You and your two friends Shelly and Lisa each have a spinner like the one below. You make up a game where everyone spins their spinner. You win if everyone gets a different color. Shelly wins if everyone gets the same color. Lisa wins if exactly two people get the same color. Analyze this game. Is it fair? If not, who has the advantage?

10. Paul's game is you toss three coins and win if you get exactly two heads. Steve's game is you toss four coins and win if you get exactly two heads. Whose game should you play to have a better chance of winning? Explain.

11. You spin the spinner below three times. You win the game if you get purple at least once. Should you play the game? Explain.

12. Deb makes up a card game. You draw two cards from a deck. If the cards are the same color (both red or both black) then she wins. If the cards are different colors (one is red and one is black) then you win. Is this game fair? Explain.

13. Rachel makes up another card game. You draw one card from a deck. If the card is a spade, a diamond, or a jack, then Rachel wins. Otherwise, you win. Is this game fair? Explain.

14. You toss two coins and roll a die. If the coins match and the die is an even number, you win. If the coins don't match and the die is greater than two, your friend wins. If anything else happens, nobody wins. Is this game fair? Explain.

15. Gerry goes on a game show similar to the one Jeff was one (from the Concept problem). This time, there are 4 doors and a car is behind one of them. Gerry will have to pick a door. Then, the host will open two of the other doors to reveal goats and ask Gerry if he wants to switch his choice. He says his strategy will be to switch when he plays the game. Is this a good strategy? Explain.

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