Category: Probability

Engaging students: Probability and odds

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Michelle Nguyen. Her topic, from Pre-Algebra: probability and odds.

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A2. How could you as a teacher create an activity or project that involves your topic?

As a teacher, I would place 100 red marbles and 25 blue marbles in a bag and have each group of students draw a marble each time from a bag for five times. After drawing a marble, the student would put the marble back and then redraw. After five times, the class would come together and the students would compare how many red marbles to how many blue marbles they have. The students will compare the ratios and guess if there are more red marbles or blue marbles in the bag given. By doing this, the students will see whether there is a big chance of drawing a red or blue marble. After doing the activities, I would ask questions that will scaffold the students into saying that there is a higher probability in picking a red marble than a blue marble because the red marbles are picked more often when compared to the blue marbles that got picked.

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D1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

With the popularity of gambling rising in the French society, mathematical methods were needed for computing chances. A popular gambler named De Mere talked to Pascal about questions about chances. Therefore, Pascal talked to his friend Fermat and they began the study of probability. The created the method called classical approach which is the probability fractions we use today. In order to verify the results of the classical approach, Fermat and Pascal used the frequency method. During this method, one would repeat a game a large number of times with the same conditions. Bernoulli wrote a book named Ars Conjectandi in 1973 to prove the classical approach and the frequency method are consistent with another one. Later on Abraham De Moive wrote a book to provide different examples of how the classical methods can be used. As time passed by, probability moved from games of chance to scientific problems. Laplace wrote a book about the theory of probability but he only considered the classical method. After the publication of this book, many mathematicians found that the classical method was unrealistic for general use and they attempted to redefine probability in terms of the frequency method. Later on, Kolmogorov developed the first rigorous approach to probability in 1933. There are still researches going on about probability in the mathematical field of measure theory.

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C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

In the movie “21” there is math problem that is similar to the popular Monty Hall problem. In the movie, a kid is given the chance to pick one out of three doors with a car in it in order to win. Once a door is chosen, the announcer will open a door without a car. Therefore, the start off is 33% of a car existing and 66% with an empty door. Since a door was open, the chance of switching your choices gives you a higher winning percentage because the one you chose at the beginning will still be 33% while switching will change your chances to 66%. This youtube video is a clip from the movie:

References:

http://www.math.wichita.edu/history/activities/prob-act.html#prob1

http://staff.ustc.edu.cn/~zwp/teach/Prob-Stat/A%20short%20history%20of%20probability.pdf

http://www.examiner.com/article/21-and-the-monty-hall-problem

 

 

“Or” / “and”

One of the formulas typically taught in mathematics is

P(A \cup B) = P(A) + P(B) - P(A \cap B)

In ordinary English, the probability that either event A or B happens is the probability of event A plus the probability of event B minus the probability that the both occur.

For example, when rolling two fair six-sided dice, the probability that at least one three appears is

P(A \cup B) = \displaystyle \frac{1}{6} + \frac{1}{6} - \frac{1}{36} = \displaystyle \frac{11}{36}.

It’s necessary to subtract something off at the end because it’s possible for the first die to be a four and simultaneously the second die to be a four.

This can be a conceptual barrier for students if it’s not directly addressed. In mathematics, the word “or” means “one or the other… or maybe both.” In the previous example, event A was “first die is a four” and event B was “second die is a four,” and it’s possible that both events could occur simultaneously.

Of course, this is different than the way we typically use “or” is spoken English. For example, in the final episode of each season of “The Bachelor,” the guy has to choose one woman or the other… and there’s no possibility of him choosing both! When a student says, “Next semester, my morning class will be history or physics,” we don’t think that there’s a possibility that the student will choose both classes… the student will choose one or the other, but not both.

In terms of mathematical logic, the word “or” in ordinary speech really is an “exclusive or.”

As I said, this isn’t a big deal for students to see, but in my opinion it’s best to directly address this subtlety rather than have students confused about which meaning of the word “or” they should be using when doing their homework.

P.S. The good news is that the word “and” means the same thing in the language of probability/logic as its meaning in ordinary speech.

The Law of Averages

Colloquially, the Law of Averages dictates that what ought to happen does happen if it happens long enough. If a gambler plays a casino for a very long time, he is almost certainly guaranteed to lose. If a weak sports team plays a stronger team in a multiple-game series, then it is almost certain to lose the series.

However, if the gambler plays in the casino for only a little while, then there is a realistic (though less than 50%) chance of coming out ahead. And a weak sports team may defeat a stronger one if only one game is played… hence the appeal of the NCAA basketball tournament and, on a larger scale, the knockout stages of the World Cup.

In my statistics class, I use a simple simple spreadsheet to illustrate that \hbox{SD}(K) = \sqrt{n p(1-p}) for a sample count, but \hbox{SD}(\hat{p}) = \displaystyle \sqrt{ \frac{p(1-p)}{n} } for a sample proportion.  Here is one image from the spreadsheet:

coinflip1

The user can change the bright green cell to be any positive integer up to 5000. This number represents the number of simulated coins that are flipped. In the above example, ten coins are flipped. Column B shows the results of the simulated coins, while column C shows a running count of the number of heads that have appeared. In the above example, 7 of the 10 flips are heads, for an observed error of +2 (two more heads than the expected number of 5) and a percentage error of 20%.

In class, I would run the spreadsheet several times, and students will see that the observed error usually is in the range of -2 to +2, and the percentage error is usually in the range of -20% to +20%.

By contrast, look what happens when the number of flips increases to a large number, like 5000.

coinflip2

There is now a larger absolute error — in this case, -28. Of course, an absolute error of that size is impossible with 10 coin flips or even 50 coin flips. However, at the same time, the percentage error is now significantly smaller (only -0.56%).

This example gives evidence for the counter-intuitive result that the absolute error grows like \sqrt{n} while the relative error decreases like \sqrt{n}.

The Monty Hall Problem

In 1990 and 1991, columnist Marilyn vos Savant (who once held the Guinness World Record for “Highest IQ”) set off a small firestorm when a reader posed the famous Monty Hall Problem to her:

Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, “Do you want to pick door #2?” Is it to your advantage to switch your choice of doors?

She gave the correct answer: it’s in your advantage to switch. This launched an avalanche of mail (this was the early 90’s, when e-mail wasn’t as popular) complaining that she gave the incorrect answer. Perhaps not surprisingly, none of the complainers actually tried the experiment for themselves. She explained her reasoning — in two different columns — and then offered a challenge:

And as this problem is of such intense interest, I’m willing to put my thinking to the test with a nationwide experiment. This is a call to math classes all across the country. Set up a probability trial exactly as outlined below and send me a chart of all the games along with a cover letter repeating just how you did it so we can make sure the methods are consistent.

One student plays the contestant, and another, the host. Label three paper cups #1, #2, and #3. While the contestant looks away, the host randomly hides a penny under a cup by throwing a die until a 1, 2, or 3 comes up. Next, the contestant randomly points to a cup by throwing a die the same way. Then the host purposely lifts up a losing cup from the two unchosen. Lastly, the contestant “stays” and lifts up his original cup to see if it covers the penny. Play “not switching” two hundred times and keep track of how often the contestant wins.

Then test the other strategy. Play the game the same way until the last instruction, at which point the contestant instead “switches” and lifts up the cup not chosen by anyone to see if it covers the penny. Play “switching” two hundred times, also.

You can read the whole exchange here: http://marilynvossavant.com/game-show-problem/

For much more information — and plenty of ways (some good, some not-so-good) of explaining this very counterintuitive result, just search “Monty Hall Problem” on either Google or YouTube.

Source: http://www.xkcd.com/1282/

In-class demo: The binomial distribution and the bell curve

Many years ago, the only available in-class technology at my university was the Microsoft Office suite — probably Office 95 or 98. This placed severe restrictions on what I could demonstrate in my statistics class, especially when I wanted to have an interactive demonstration of how the binomial distribution gets closer and closer to the bell curve as the number of trials increases (as long as both np and n(1-p) are also decently large).

The spreadsheet in the link below is what I developed. It shows

  • The probability histogram of the binomial distribution for n \le 150
  • The bell curve with mean \mu = np and standard deviation \sigma = \sqrt{np(1-p)}
  • Also, the minimum and maximum values on the x-axis can be adjusted. For example, if n = 100 and p = 0.01, it doesn’t make much sense to show the full histogram; it suffices to have a maximum value around 5 or so.

In class, I take about 3-5 minutes to demonstrate the following ideas with the spreadsheet:

  • If n is large and both np and n(1-p) are greater than 10, then the normal curve provides a decent approximation to the binomial distribution.
  • The probability distribution provides exact answers to probability questions, while the normal curve provides approximate answers.
  • If n is small, then the normal approximation isn’t very good.
  • If n is large but p is small, then the normal approximation isn’t very good. I’ll say in words that there is a decent approximation under this limit, namely the Poisson distribution, but (for a class in statistics) I won’t say much more than that.

Doubtlessly, there are equally good pedagogical tools for this purpose. However, at the time I was limited to Microsoft products, and it took me untold hours to figure out how to get Excel to draw the probability histogram. So I continue to use this spreadsheet in my classes to demonstrate to students this application of the Central Limit Theorem.

Excel spreadhseet: binomial.xlsx

Mathematics and The Price Is Right

I just read a very entertaining article on the use of game theory for improving contestants’ odds of winning the various games on the long-running television game show “The Price Is Right.” Quoting from the article:

On a crisp November day eight years ago, I took the only sick day of my four years of high school. I was laid up with an awful fever, and annoyed that I was missing geometry class, which at the time was the highlight of my day. I flipped on the television in the hope of finding some distraction from my woes, but what I found only made me more upset: A contestant named Margie who was in the process of completely bungling her six chances of making it out of Contestants’ Row on The Price is Right.

Many contestants fail to win anything on The Price is Right, of course. But as I watched the venerable game show that morning, it quickly became clear to me that most contestants haven’t thought through the structure of the game they’re so excited to be playing. It didn’t bother me that Margie didn’t know how much a stainless steel oven range costs; that’s a relatively obscure fact. It bothered me, as a budding mathematician, that she failed to use basic game theory to help her advance. If she’d applied a few principles of game theory—the science of decision-making used by economists and generals—she could have planted a big kiss on Bob Barker’s cheek, and maybe have gone home with … a new car! Instead, she went home empty-handed…

To help future contestants avoid Margie’s fate, I decided to make a handy cheat sheet explaining how to win The Price Is Right—not just the Contestants’ Row segment, but all of its many pricing games. This guide, which conveniently fits on the front and back of an 8.5-by-11-inch piece of paper, does not rely on the prices of items.

The full article can be found at http://www.slate.com/articles/arts/culturebox/2013/11/winning_the_price_is_right_strategies_for_contestants_row_plinko_and_the.html.