Unit IV
Chances Are

Algebraic models for patterns of change in related variables make very specific and well-defined predictions--if x = 3, then 5(2)x = 40. But most situations in the real world are not so neatly and certainly predictable. There is an element of random variation in almost every aspect of the physical, biological, economic, and social world in which we live. The principal branches of mathematics that offer reasoning tools for dealing with randomness are statistics and probability. The investigations of this unit are designed to illustrate several of the main concepts and techniques of elementary probability and the way those mathematical tools can be applied to amusing and important problems.

In common language the word random suggests to most people notions of unpredictability or absence of any pattern or regularity. From the perspective of mathematical probability and statistics, random events are viewed quite differently. A random phenomenon is an activity for which outcomes of individual cannot be predicted with certainty but results of many repetitions fall into quite predictable patterns.

For any specific possible outcome of a random activity, the fraction comparing the number of occurrences of that outcome to the total number of trials of the activity is called the probability of the outcome. For example, in the activity of rolling a single fair die, there is no way to predict the outcome of any one roll. However, out of a large number of rolls, we can reasonably expect about 1/6 to show five dots on the up face. So it is common to say that the probability of getting "5" on one roll of a fair die is 1/6.

There are three basic ways to calculate probabilities for random phenomena: Perform the actual random activity many times and keep a record of outcomes; design a simulation of the real activity, repeat it many times, and keep a record of outcomes; or analyze the random activity mentally and use that reasoning to estimate probabilities. The problems of this unit provide opportunities to learn how both simulation and analysis can help to study random events through use of mathematical models.

4.1 Let's Make a Deal

One of the best known and most intriguing elementary probability problems is based on a television game show from the 1970's called Let's Make a Deal. In the show the host, Monty Hall, would ask a contestant to pick one of three doors to open, in hopes of finding a very large prize. As soon as the contestant made his or her selection, Monty would open one of the two remaining doors--one which he was certain did not hide the prize. Then he would offer the contestant a chance to change his or her selection.

The question always was, "Does it make sense to change one's pick, should one stay with the original pick, or does it make no difference at all in the probability of winning?"

Think About This Situation

What would your choice be? How could you go about studying this situation to get some ideas about the probability of winning in each case?

You could explore the Monty Hall problem by locating tapes of the show and studying the data of people who do and don't switch. That's a bit complicated! You could also run a simulation of the show with your group partners.

1. How could you simulate the Let's Make a Deal show and its door opening game so that you'd get data that would help estimate the probability of winning with two different strategies? When you have a convincing plan, test it out and see what it says about the two options--switch or not switch.

Simulation is an extremely adaptable strategy for making sensible estimates of probabilities. Suitable simulations can make use of physical devices like spinners or chips in a bag and so on. Or they can use random numbers in a table or as presented by a calculator random number generator.

On the TI-82 graphing calculator, you can generate random numbers selected from any range of numbers (whole numbers or decimals) with a few simple commands. For example,

Press MATH and on the resulting menu choose PRB and rand
Your screen will show "rand"

Pressing ENTER repeatedly will give a sequence of random numbers
chosen from between 0 and 1.

To get other sorts of random number lists, you can try these variations:

When you get "rand" on your screen, type a number after the "rand" like "rand5" and then repeatedly press ENTER to get a sequence of random numbers between 0 and 5.

To get only random integers from a set like 0, 1, 2, 3, 4, 5, 6, 7, 8 you can first select MATH Num 2 to get "iPart" on your screen. Then produce "rand9" to get a random set of integers from 0 to 8.

2. One way to simulate the Monty Hall problem is as follows:

* Get two calculators to produce random numbers from the set of possibilities 0, 1, or 2.

* Consider one random digit to be the door behind which the prize is placed (we'd want that to be chosen randomly).

* Consider the second random digit to be the player's original guess.

* Determine which door would be opened by Monty first and then record what the player would get (win or lose) in case he/she switched and in case he/she did not switch.

* Investigate the long-run relative frequency of winning and losing if you adopt a switching strategy and if you adopt a non-switching strategy.

Try this to make an estimate for the probability of winning in the long run from each strategy.

3. If you roll a fair die many times, you'd expect each number outcome to occur roughly an equal number of times. The question is, how many times do you have to repeat the experiment of rolling the die in order for the "law of averages" to even things out.

(a) Roll a fair die many times, recording the results in groups of 6 rolls in a table like this that shows the fraction of "fives" at various stages in the process:
Total tosses    6     12    18     24     30     36     42     48     54     60     
Number of 5's                                                                       
Fraction of                                                                         
(b) Design a random number simulation for die rolling and carry it out recording the results in a table like that for (a).

(c) Compare the results in (a) and (b) and comment on how they do or do not illustrate the mathematical meaning of random as "uncertain result of any single trial, but pattern in many repetitions."

4. Simulations are useful strategies for estimating probabilities in real-life serious problems as well. For example, consider the problem of population control in China. With a population of about 1.2 billion and a land area of 3,700,000 square miles, it is much more densely populated than the United States where the population of about 260 million is spread (albeit not evenly) over an area of about 3,600,000 square miles.

Since it is important in Chinese culture (especially in rural areas) to have at least one male child, one might reasonably ask the probability of getting small and large families if the official policy allows families to have more and more children until they get the first male heir.

Assuming that male and female children are equally likely on any one birth, what simulation will be useful to estimate the probabilities of various family sizes under the proposed policy.

(a) How could you simulate births of male or female children by flipping a coin?

(b) How could you do the simulation by rolling a fair die?

(c) How could you do the simulation by using calculator generated random numbers?

(d) How could you do the simulation by selecting a bead from a collection of red and white beads?

(e) Carry out at least two different simulations to estimate the probabilities that the first male child will occur on births 1, 2, 3, 4, 5, 6, 7, or 8.

5. Sports statistics give a basis for estimating probabilities of many different outcomes in games from football and basketball to baseball and tennis. For example, suppose that a basketball player has made 70 of 100 free throw attempts in a season.

If that player comes to the free throw line for two shots, what are the probabilities that the player makes 0, 1, or 2 of the shots?

(a) How could you simulate free throw shooting using a collection of beads in a bag--some white and some red?

(b) How could you do the simulation with a spinner?

(c) How could you do the simulation by using calculator generated random numbers?

(d) Carry out at least two different simulations to estimate the probabilities that the shooter makes 0, 1, or 2 shots.

6. Jury selection in the American criminal justice system is a sensitive issue. The constitution guarantees everyone accused of a crime a trial by a jury of his/her peers. When a jury does not seem to reflect the distribution of gender or race in the total population, there is often an appeal of the decisions by that jury.

Traditional juries have 12 members, but for some matters it is now common to have juries of only 6. Suppose that a jury of 6 in some case turns out to have no female members.

Assuming that females and males are essentially equal in number in the population, what is the probability of getting an all male jury, if the selection was really gender-neutral?

(a) How could you simulate selection of male or female jury members by flipping a

(b) How could you do the simulation by rolling a fair die?

(c) How could you do the simulation by using calculator generated random numbers?

(d) How could you do the simulation by selecting a bead from a collection of red and white beads?

(e) Carry out at least two different simulations to estimate the probabilities that the jury of six will have 0, 1, 2, 3, 4, 5, or 6 females.

7. Suppose that on a multiple choice test each question offers 4 options, only one of which is correct.

(a) What is the probability of getting any one question correct by pure guessing?

(b) On a test of 5 such questions, what is the probability of getting at least 3 correct by pure guessing?

8. When companies make large purchases of manufactured goods, they often test samples from a shipment before deciding whether to accept or reject the order. If their tests reveal too many faulty items, they might reason that the shipment is badly flawed and reject the shipment. At the same time, it is often not possible to fully test each item in a shipment because the product (e.g. fireworks) must be destroyed in the testing!

Suppose that a shipment of 60 graphing calculators actually contained 10 defective machines. However, because of the trouble to open, install batteries, and fully test a calculator, the retailer decides to test only five from the order.

What is the probability of finding at least one of the 10 defective machines in the testing strategy?

(a) Devise a simulation strategy with random numbers for estimating the probability that at least one of the 10 defective machines will be found in a test of 5.

(b) Devise a second simulation strategy using some physical materials like beads in a bag, a spinner, or a die.

(c) Carry out the two simulations to estimate answers to the original question.

9. Some contests and other advertising gimmicks for products involve a situation like this: Packages of some product contain various "prizes". The object is to collect all of the different possible prizes.

Suppose that a soft-drink company places 6 different "prize" symbols at random on the bottom of cans of their product. When you collect cans with all 6 different symbols, you can send them to a distributor for a $25 reward.

How many cans do you think you'd have to buy to get a full set of 6 prizes?

(a) Devise a random number simulation of that can be used to estimate the probability that you will get all 6 prize symbols by buying only 6 cans of the soft drink. Carry out your simulation until you feel confident of your estimate.

(b) Use the data from trials of (a) to estimate the probability that it will take 7, 8, 9, 10, 11, or 12 cans to get the full set.

(c) What other kind of simulation could you use to estimate the probabilities called for in (a) and (b)?

4.2 Roller Derby

Designing a good simulation requires careful analysis of the probability situation it models. But then it takes a lot of patience and record-keeping to get data that can be turned into probability estimates. The following game can be played more skillfully if you do some thinking about the probabilities involved:

* The game, called Roller Derby, is a contest between two individuals or teams. Each team will need 12 markers or beads and 12 small cups. Together the teams need a pair of fair dice.

* The paper cups should be numbered 1, 2, 3, ..., 12 and the 12 markers or beads distributed into the 12 cups in some way (not necessarily all in one or one in each).

* The players or teams alternate turns rolling the pair of dice. For whatever sum occurs, each team can remove one marker or bead from the cup whose number matches the sum of the faces showing on the dice. If there are no markers left in the cup whose number occurs, nothing occurs.

* The winner is the individual or team who clears all cups first.

1. Make a plan for distributing the markers or beads into the cups, record your plan, and then play the game against an opponent. Adjust your strategy after each game, if you think it is advisable.

(a) What distribution strategies seem smartest?

(b) How did you arrive at your strategy that seemed best?

There are various plausible strategies for making smart moves in the Roller Derby game. It seems likely that any such strategy would begin with a listing of the possible outcomes of any roll of two dice. For example, one list would be:

Die Toss Outcomes: 1,2,3,4,5,6,7,8,9,10,11,12

The first thing you notice when you think about things a little is that one should never put a marker or bead in the cup numbered 1, because that is an impossible outcome. It has probability 0 of occurring.

The next thing you might notice when you think about the situation (perhaps after you've played the game once or twice) is that the possible outcomes are not equally likely. To get a better idea of what is likely to happen, it would be sensible to make a list of possible outcomes in which every listed possibility is equally likely. For the toss of two dice, that involves ordered pairs of outcomes from the individual dice like this:

Die Toss Outcomes:

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

From this listing of all the equally likely outcomes, there are some logical probability estimates for the various possible sums. For instance, of the 36 ordered pairs only 1 gives a sum of 2. So it seems reasonable to estimate the probability of getting a 2 to be 1/6 . Similar reasoning leads to the estimate of 6/36 or 1/6 for the probability of getting a sum of 7 on the two dice.

2. Continue the reasoning about equally likely outcomes to complete the following table so it shows a plausible estimate of the probability of any possible sum of the two dice used in Roller Derby:
Possible Sum of Dice     2    3      4   5    6    7    8    9    10   11   12   
Probability Estimate                                                             
(a) How would you use the information in this table to devise an optimal strategy for playing Roller Derby?

(b) How confident are you that you'd always win at Roller Derby if you played the optimal strategy you proposed in (a)?

The table you completed in (2) is one example of what is called a theoretical probability model. It has two central features: First is a list of possible outcomes of the random activity being studied; second is an assignment of probabilities to each of the conceivable outcomes.

In this, as in many similar situations, a good probability models allows you to make a variety of probability predictions.

3. Using the model you developed in (2) estimate the probabilities of these random events:

(a) Sum of the dice is a number less than 5.

(b) Sum of the dice is an odd number.

(c) Sum of the dice is a multiple of 3.

(d) Sum of the dice is a number greater than or equal to 5.

(e) Sum of the dice is a number between 5 and 8.

4. Next, consider the game in which two dice are rolled and the numbers showing are multiplied instead of summed.

(a) Make a probability model that could be used in estimating likelihood of various outcomes in this game.
Possible Product of                                                              
Probability Estimate                                                             
(b) Find the probabilities of the events described in (3) with the word sum replaced by the word product.

As you worked on devising good theoretical probability models for the die-tossing activities, you probably found it very convenient to refer to the full list of possible outcomes in this matrix:

Die Toss Outcomes:

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

This set of outcomes shows all 36 possible results from tossing two dice. Furthermore, all of those outcomes are equally likely. That basic outcome set can be used to calculate theoretical probability estimates for many other specific events. The probability of any of the equally likely outcomes in this list is 1/6. Can you see why that is reasonable?

5. Use the set of equally likely outcomes to estimate probabilities for the following specific events:

(a) The second die is greater than the first.

(b) The second die is a multiple of the first.

(c) The second die is a factor of the first.

(d) The product of the two dice is an odd number.

6. In some probability situations you are required to make decisions based on partial evidence from a random event. For instance:

(a) Find the probability that the sum of two dice is greater than 8, given the information that the first die is a 5.

(b) Find the probability that the product of two dice is an odd number, given the information that the first die is 4.

(c) Find the probability that the second die is less than the first, given the information that the first die is an odd number.

(d) Find the probability that the second die is less than the first, given the information that the first die is an even number.

Building and using a theoretical probability model is a strategy that is useful in a variety of situations that don't involve die tossing! See if you can construct a suitable set of outcomes and assignment of probabilities to solve the following problems:

7. Promotional contests and lotteries of many kinds offer "scratch games" like this simple example:

A Scratch Five card has 5 disks with erasable ink covering messages. Two of the disks are Win $5 messages; the other three are You Lose messages. The object is to scratch off the covers of two disks to get the two winners.

(a) What is the probability of winning the $5 prize in this game?

(b) What probability model supports your estimate?

8. Can you curl your tongue? Some people can do it easily; others cannot do it, despite valiant efforts. It turns out that tongue curling ability is determined by a special gene form called an allele. Every person has two tongue-curling alleles--one inherited from your mother and one from your father.

Of the two kinds of tongue-curling alleles, one is called dominant and the other recessive. Using T for dominant and t for recessive, a person will be able to curl his or her tongue if and only if he/she inherits at least one T allele.

(a) If you can curl your tongue, what can you infer about the tongue-curling alleles of your biological parents?

(b) If both of your biological parents can curl their tongues, what does that imply about your tongue-curling inheritance?

(c) What is the probability that a child will be able to curl his or her tongue in these cases of biological parent tongue curling alleles:

(i) Father TT and Mother Tt

(ii) Father Tt and Mother Tt

(iii) Father Tt and Mother tt

(iv) Father tt and Mother tt

4.3 Clever Counting

Among the hundreds of state and local lottery and casino gambling games are many different odds of winning. With most such games the number of possible outcomes is very large, so it is not a simple matter to list all possible outcomes and then calculate probabilities for various combinations. For example, in a common Power Ball game each player chooses four numbers from 1, 2, 3, ... 40. Then you choose a fifth power ball number from the original list of 1, 2, 3, ..., 40. The winning combination of four numbers and the power ball is chosen by some random process run by the lottery operators. If your four numbers and power ball match the winning numbers you can win a very large prize. Various other combinations sometimes earn smaller prizes.

The challenge in figuring odds for these complex games is to be able to count the number of possible outcomes without actually writing out a full list. The following problems should suggest some strategies for such counting.

1. Automobile license plates in nearly every state have some combination of letters and numbers. In Maryland the standard plate is three letters followed by three numbers. For example, ABC 123 or ACB 364 or XAX 212 or BBB 999, etc.

(a) How many different license plates could be made up with a single letter and three numbers--like A 987 or B 788 etc.?

(b) How many different license plates could be made up with two letters and three numbers--like AA 997 or AB 782 or BC 666?

(c) How many different license plates could be made up with three letters and three numbers?

2. Every radio and television station east of the Mississippi River has a set of call letters beginning with the letter W and 2 or 3 additional letters. For example, in Baltimore there is WMAR television and WBAL radio. In the Washington, DC area there are WRC television, WPGC radio, and so on.

(a) How many different station call letter combinations can be made up in the form
W __ __ ? How about W __ __ __?

(b) Consider the problem in (a) with the new condition that no letters may be repeated in a station call letter name? For example, W A A B is not allowed.

3. How many different "standard" license plates (3 letters followed by 3 numbers) can be composed in which no letter or numeral is repeated?

(a) Consider first the simpler case of only three letters with no repetition of letters.

(b) Consider next the simpler case of only three numerals used with no repetition.

(c) Then put together the answers of (a) and (b) to answer the original question.

4. Consider next the question of how many ways one can arrange the letters of a radio or television station call signal in different orders. For example, the letters in WPGC can be reordered as WPCG, WCPG, WCGP, etc.

(a) In how many orders can the letters A, B, C be arranged?

(b) In how many orders can the letters A, B, C, and D be arranged?

(c) In how many orders can the letters A, B, C, D, and E be arranged?

5. To play the Power Ball game you must first select four different numbers from among
1, 2, 3, ..., 40. Then you select a fifth power ball from the same list (The power ball can be a repeat of one of the original four.).

(a) How many different ordered selections of four numbers are possible?

(b) How many different unordered selections of four numbers are possible?

(c) What is the probability of getting all four choices to match the winning number if you make your selection totally at random (as the winning number presumably is)?

(d) How many different selections of four numbers and one power ball are possible?

(e) What is the probability of winning the grand power ball prize if you purchase a single ticket?

Conclusions and Connections

The problems in this short sample of probability modeling only begin to illustrate the ideas and techniques available for thinking about random phenomena. The key notions to keep in mind are answers to the following questions:

1. What is a random event or phenomenon, from a mathematical perspective?

2. What does the probability of such a random phenomenon tell about it?

3. How can one reason about probabilities using:

(a) Simulation models?

(b) Theoretical probability models?

4. How are probability models similar to and different from algebraic and
geometric models used in earlier investigations?