Setting Up Binomial and Poisson Probability Problems

I have uploaded 2 videos to YouTube that go over how to set up binomial and Poisson probability problems. (There are 8 binomial problems, and 6 Poisson problems.) I go over the steps for identifying the problems, as well as give the correct answers. If you’d like a copy of the actual problems, just drop me a line.

Binomial: http://www.youtube.com/watch?v=jLAePWjEZYE
Poisson: http://www.youtube.com/watch?v=GupBzWFL-KY

– George

I am a math instructor at College of the Sequoias in Visalia, CA. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.

Letting Your Students Think

When teaching a new topic, we have to resist the temptation to just tell our students how to solve a problem and let them think for themselves. This helps our students to develop their intuition and their problem solving skills.We also have to keep in mind that there is more than one way to solve a problem.

Example 1 – Intro Stats

The example that jumps to mind is from a statistics class when students need to find P(a < z < b)  using the normal distribution. In class we have already learned how to find P(z < a) using the z-table and P(z > a) using the complement of the area from the z-table. I draw a bell curve and ask my students to think about how they can find this shaded “middle” area. For me, the most efficient approach is to find P(z < b) and then subtract P(z < a).

When I ask my students for their ideas, someone will always say to

1. Find P(z < a) using the z-table.
2. Find P(z > b) using the complement of the value found in the z-table.
3. Add these two areas together, and subtract their sum from 1.

To me, this approach is not as efficient as the one I laid out, but I love it when my students suggest it. First, it shows me that my students have considered the work we have done so far and applied it to this new problem. Second, it allows me to explain why the other approach is more efficient.

Example 2 – Series

Last night my son (HS Sophomore) was working on a midterm review. One problem asked him to find the sum 3 + 1/3 + 1/9 + 1/27 + … This sum is not quite in the form of a geometric series, but it’s close. The approach that jumped out to me was to rewrite 3 as 2+1, resulting in 2 + (1 + 1/3 + 1/9 + 1/27 + …). I could then use a formula to find the sum in parentheses and add 2 to this result.

My son had a different idea – why not add in a 1 after the 3 to make the series a geometric series, find the sum, and then subtract 1? What a great idea! There are many times in math where we use the add & subtract a number trick, it was nice to see him come up with that on his own.

I then asked him if he could find a third way of finding the sum. After a little thought, he realized that 1/3 + 1/9 + 1/27 + … was itself a geometric series and that he could find that sum and add 3 to the result.

(Side note: The formula for the limit of a geometric series with initial term a1 and common ratio r is given by r=a1/(1-r). The sum for the given problem is 3.5.)

I think my son gained a lot more than the solution to the one particular problem – he developed mathematical intuition, increased his understanding about geometric series, and further developed his critical thinking skills. Win – Win – Win!

Summary

If we tell our students to simply do what we do, what will they do when they run into a problem they have never seen before and we are not there to tell them what to do? Encouraging students to think, and to develop strategies on their own, prepares them to be a lifelong problem solver. Isn’t that what we want? Let your students think.

-George

I am a math instructor at College of the Sequoias in Visalia, CA.  If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.

Statistics – Building Intuition About The Central Limit Theorem Using Excel

Here’s how I introduced the Central Limit Theorem in class on Thursday. We took an exam on binomial, Poisson, and normal probabilities, so I began with a review question – Find the probability that a randomly selected adult has an IQ of 97 or below (mean: 100, std dev: 15). The students worked through the problem, and found that P(z < -0.2) = 0.4207.

I then asked the class whether it would be more likely or less likely that a group of 100 people would have a mean IQ of 97 or lower. My students weren’t really sure how to think about this. I reminded them of the law of large numbers, and a student commented that as the sample size increased the mean should be drawn towards the population mean (100), so she said it would be less likely that the mean would be below 100.

Before introducing the formulas from the Central Limit Theorem, I opened up Excel for some simulation work. I generated a column with 100 random values in it, sampled from a normal distribution with a mean of 100 and standard deviation of 15. This can be done using the data analysis toolpak. I calculated the mean and standard deviation for this sample – the mean was around 99 and the standard deviation was around 14.5. We talked about these values making sense, the mean and standard deviation should be close to those of the population. We sorted the values and saw that 40 of them were 97 or lower, which was close to the 42% we expected.

I then used the data analysis toolpak to generate 100 samples in the same fashion. We looked at the mean of each sample and noticed that they seemed to be centered around 100, but did not vary as much as the individual values did. We sorted these means and noticed that only 2 of the sample means were 97 or lower, much lower than the number of individual values in that category. We calculated the mean of the sample means (99.97) and standard deviation of the sample means (around 1.5), showing that the mean was roughly equal to the population mean and that the standard deviation was significantly lower than the population standard deviation. I finished by creating a histogram of these sample means, and we noticed that the shape was roughly bell shaped.

At this point I introduced the Central Limit Theorem and the formulas. My students understood the Central Limit Theorem before the formulas were ever presented, and that should help to understand confidence intervals conceptually.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. If there’s a particular statistics topic you’d like me to address, or if you have a question or a comment, please let me know. You can also reach me through the contact page on my website – http://georgewoodbury.com.

StatCrunch – Probability Calculators

My statistics classes have just finished the unit on probability distributions – binomial, Poisson, and normal. This is the first semester using StatCrunch in this course, so I thought I’d put together a short “how to” article on how to use StatCrunch with these three distributions.

Binomial Distribution

The binomial calculator in StatCrunch can be accessed through Stat > Calculators > Binomial. The dialog box asks you to enter n, p, and x. You also enter an inequality or equal sign: <, <, >, >, or =. The calculator calculates the probability and displays a probability histogram. Values of x in your interval appear in red in the probability histogram.

One slight problem arises when trying to determine P(a < x < b). You must first calculate P(x < b), and from this result subtract P(x < a). Alternatively, you could find P(x = a), P(x = a + 1), …, P(x = b) and total these results.

Poisson Distribution

The Poisson calculator in StatCrunch can be accessed through Stat > Calculators > Poisson. The dialog box asks you to enter the mean and x. You also enter an inequality or equal sign: <, <, >, >, or =. The calculator calculates the probability and displays a probability histogram. Values of x in your interval appear in red in the probability histogram.

As with the binomial calculator, you cannot directly calculate P(a < x < b) with the Poisson calculator. You must first calculate P(x < b), and from this result subtract P(x < a). Alternatively, you could find P(x = a), P(x = a + 1), …, P(x = b) and total these results.

Normal Distribution

The normal calculator in StatCrunch can be accessed through Stat > Calculators > Normal. The dialog box asks you to enter μ and σ. These are defaulted to 0 and 1 respectively, the parameters for the standard normal (Z) distribution. These can be changed to work with any normal distribution. You also enter an inequality sign: < or >. The calculator calculates the probability and displays a normal curve with the appropriate area shaded.

Should you need to calculate P(a < x < b) you must first calculate P(x < b) and from this result subtract P(x < a).

The normal calculator also handles what I call “reverse normal” problems in which you are given a probability and asked to find the value of the random variable associated with it. Simply enter values for the mean and standard deviation, as well as the probability and inequality (left tailed or right tailed). When you click the Compute button, StatCrunch gives you the value of the variable that fits the given situation.

Ease Of Use

My students found these calculators very intuitive to use. When I presented them in class I let the students tell me how to enter values in the dialog boxes. They were able to do so with no difficulty.

Conclusion

These calculators are powerful and very easy to use. Students still need to understand each distribution, as well as how to tell the difference between them, in order to be successful with StatCrunch. Now, if I could only come up with a way to test them using StatCrunch even though our class does not meet in a computer lab. Any suggestions?

-George

I am a mathematics instructor at College of the Sequoias in Visalia, CA. I have decided to add technology related articles, including articles based on StatCrunch, to my Thursday blog lineup. Let me know if there are other topics you’d like me to cover by leaving a comment or by reaching me through the contact page on my website.

Using StatCrunch To Present Normal Approximation To The Binomial Distribution

Today I introduced the normal approximation to the binomial distribution to my class. I used StatCrunch to help with the presentation and it went really well.

Beginning Example

Since it has been a week or so since we were working with the binomial distribution, I put the following problem on the board to discuss.

Example 1: 60% of students at our college are female. If we randomly select 10 students, find the probability that 7 are female.

Students identified that it was a binomial problem, and we talked about how to identify a problem as a binomial problem. We then talked about why using the binomial formula was an effective strategy for solving this problem. I then edited the problem as shown.

Example 2: 60% of students at our college are female. If we randomly select 10 students, find the probability that at least 7 are female.

Students commented that by rewriting 7 as “at least 7” made the problem a little more difficult, but we could then use the binomial tables and add the probabilities associated with x=7 through x=10. So I edited the problem again.

Example 3: 58% of students at our college are female. If we randomly select 10 students, find the probability that at least 7 are female.

Changing p from 0.6 to 0.58 meant that the binomial tables were no longer an option. The students understood that they would need to use the binomial formula for x=7 through x=10. Although it was a little inconvenient, the problem could still be handled in a relatively easy manner. That’s when I changed the problem one more time.

Example 4: 58% of students at our college are female. If we randomly select 100 students, find the probability that at least 70 are female.

OK, now we have a problem. We cannot use the binomial tables, and using the formula would require us to use 31 different values of x. Using the complement would require us to use 70 different values of x.

Enter StatCrunch

I opened up StatCrunch and went to the binomial calculator (Stat > Calculators > Binomial).

We could see that technology could give us a solution rather quickly. The probability is 0.0090. But we also saw something that was very valuable. My students saw that the probability histogram was approximately normal. They were thinking that it was too much of a coincidence. I commented that it was too bad we didn’t know the mean or standard deviation for this distribution, and a student quickly reminded me that we did have a formula for those.

Continuity Correction

One stumbling block for past students is how to apply a continuity correction, or even understanding why. With the probability histogram still on the board I could clearly point out the gaps between the possible discrete values of x, and marked how we split those gaps in half. I also was able to explain why we made our adjustment downward in this example because the histogram was already red for every value beginning at 70, but we needed to drop down to 69.5 (the lower limit for x = 70).

Normal Approximation

We walked through the steps and students saw that we could get a reasonably good approximation of the probability without having to use the formula 31 times. The students appreciated this efficient estimation, and truly seemed to understand what was going on rather than simply mimicking my work.

Conditions To Apply The Normal Approximation

The condition to apply the normal approximation is given in our textbook as np(1-p)>10. Using StatCrunch I was able to show how the probability histogram becomes more bell-shaped as the value of n increases.

n=10

n=20  

n=50

n=100

n=1000

Conclusion

All in all, a pretty productive day with a classroom full of students who truly understood.

-George

I am a mathematics instructor at College of the Sequoias in Visalia, CA. I blog about general teaching ideas every Wednesday. Let me know if there are other topics you’d like me to cover. You can email suggestions through the contact page on my website.

Statistics – My New Approach Continues

Today I covered discrete probability distributions in my statistics course. In the past, when discussing the mean and standard deviation of a probability distribution, I’d present the formulas and give a quick idea of what the mean and standard deviation represented. Looking back, I know my students most likely didn’t understand what they were doing, they just did it.

I’m trying to make things a little more conceptual this semester, with a focus on understanding. Here’s an example of my new approach.

I started by putting a table on the board for the probabilities of having 0, 1, 2, or 3 girls in a family with 3 children. One of students said “Mr. Woodbury, I never understand it when I read in the paper about the typical family having, like, 2.3 kids. What is that all about?” I could not have scripted it any better. I replied that no family has 2.3 children, but if we examined all families, the average number of children would be 2.3.

At this point I asked her, “If you took all the families with 3 children, how many girls would there be per family?” I brought up Microsoft Excel and typed “=RANDBETWEEN(0,1)” in cell A3, and copied the same expression to cells B3 & C3. I then copied these cells down to row 1002. “Do you know what I’ve done? I’ve simulated the genders of 3 children for 1000 families. The 0 represents that the child is a boy and a 1 represents that the child is a girl.” I typed “=SUM(A3:C3)” in cell D3 to accumulate the number of girls for each family. “See, the first family was BGB, so there is 1 girl. The second family was GGB, so there are 2 girls. And so on.”

To calculate the mean number of girls I typed “=AVERAGE(D3:D1002)” in cell D1. To calculate the standard deviation for the number of girls I typed “=STDEV(D3:d1002)” in cell D2. At this point they understood that the mean of a probability distribution was similar to the mean of a set of data, and that standard deviation was similar as well. They saw that the mean was close to 1.5, which made sense to them. By pressing the F9 key, I randomly generated 1000 families and again the mean was close to 1.5. I then started pressing the F9 key rapidly and noticed that the values danced all around 1.5. They also noticed that the standard deviations danced around 0.87.

When we went through the formulas they understood what we were trying to find, and they understood that the answers made sense.

Students are less likely to think that a subject is useless when you present the material in this fashion. Thinking is greater than mimicking. My students are more involved this semester, and seem to be doing much better as well.

I’d love to hear your comments. If there are activities or assignments that you are using to increase understanding or make connections, share them in the comment area, or send them to me through the contact page on my website – georgewoodbury.com .

-George

I am a mathematics instructor at College of the Sequoias in Visalia, CA. I blog about general teaching ideas every Wednesday. Let me know if there are other topics you’d like me to cover. You can email suggestions through the contact page on my website.

Recreational Math – How Much To Bet In Texas Hold ‘Em

In this post I will show how a little probability and a little elementary algebra can be used to determine how large of a bet you should make while playing No Limit Texas Hold ‘Em. (For a quick explanation of the rules, check out this video by Phil Gordon.)

Set Up

You have a high pair, let’s say it is a pair of Aces (A♣, A♦). There are 400 chips in the pot, and you and only one other player remain in the hand.

The flop hits the table containing no Aces, but it does contain two hearts: 10♥, 6♦, 2♥. You know there is a decent chance that your opponent holds two hearts in his hand, because he likes to play suited cards.

The question is – How much should you bet? Paraphrasing the “Fundamental Theorem of Poker”, you should bet enough so that your opponent is making a mistake if he calls.

What Are The Odds Against Your Opponent Completing Their Flush?

From your opponent’s point of view, he has seen 5 cards and 4 of them are hearts. That means that there are still 9 hearts in the deck that would complete the flush, and the other 38 would not. So the odds against the flush are 38:9, or approximately 4.2:1.

How Much Should You Bet?

If the opponent is being offered less than 4.2 to 1 on his wager, he is making a mistake to call. The goal is to make a bet (x) large enough so the ratio of the pot (400 + x) to the amount he would have to add (x) is less than 4.2. Here comes the algebra. We will determine the value of x for which (400 + x)/x = 4.2, this will give us the mathematically fair wager.

(400 + x)/x = 4.2

400 + x = 4.2x      (Note that x cannot equal 0 in this problem.)

400 = 3.2x

x = 400/3.2 = 125

A bet of $125 creates a pot of $525, offering your opponent 525:125 (or 4.2:1) odds for calling.

Any wager greater than $125, will offer your opponents insufficient odds to call. For example, a bet of $200 increases the pot to $600 and offers your opponent only 3:1 odds.

The trick is to size your bet as large as you can while still getting your opponent to call.

And If There Is No Heart On The Flop …?

You get another chance to force a bad bet. Suppose you bet $200 and your opponent called, the pot is now $800. The odds against completing the flush on the next card are now 37:9 or approximately 4.1:1. The same strategy still applies. In general, with a pot size of P, the bet size x should satisfy the following inequality:

(P + x)/x < 37/9

We find that if x > 9 P / 28, that our opponent will make a mistake to call the bet. If the pot is $800, this equates to a bet larger than$257.14. Again, the important idea is to make as large of a bet as you can that you feel will still be called.

“May all your cards be live and your pots be monsters!”

-George

I am a mathematics instructor at College of the Sequoias in Visalia, CA. Each Friday my blog contains an article on recreational mathematics. Let me know if there are other topics you’d like me to address. You can reach me hrough the contact page on my website – http://georgewoodbury.com.

Recreational Math – Baseball (Top of 9th, Man on 1st, 0 Out)

Baseball was the game of my youth. I spent countless summer days playing wiffle ball from dusk until dawn, taking a short break for a sandwich, the ice cream man, and the Del’s frozen lemonade truck. (I can still taste the Del’s! in my mind!) Growing up in Rhode Island, I was also a die hard Red Sox fan. When I woke up, I read the sports section recap while I ate and closed my day listening to the game on the radio.

Baseball is also the game of my adulthood, but for different reasons. First, the Red Sox have managed to win a pair of World Series titles. Winning is good. Second, there are so many applications of math, probability, and statistics in baseball. Some credit the use of sabermetrics with helping the Sox to win, both by the front office (building the team) and the manager (making game decisions based on data and probabilities).

No Outs, Top of the Ninth, Man on First – What To Do?

In today’s article I examine an issue of contention between baseball purists and sabermetricians, tying it to probability that can be used in an Intro Stats course. Here’s the situation – the game is tied heading to the top of the 9th inning. The visiting team manages to get its leadoff hitter on first base. Should he try to steal second base? Should the next hitter try a sacrifice bunt to advance the runner to “scoring position” at second base? Let’s analyze the options.

Win Expectancy

I am basing my analysis on “Win Expectancy”. Win Expectancy is the probability of a team winning a game based on that game’s “state” – inning, outs, baserunners. The website I am using (http://winexp.walkoffbalk.com/expectancy/search) to calculate these probabilities coded games from 1997 through 2006, counts how many games were in that state, and the percentage of times that the home team went on to win the game.

As the inning starts, the visiting team has a 47.8% chance of winning the game. (If the visiting team fails to score, that chance drops to 34.4%.) If the first batter reaches 1st base safely, the win expectancy rises by nearly 9% to 56.7%.

Should the Runner Try to Steal 2nd Base?

If the runner successfully steals 2nd base, the chances for victory jump to 65.7%. If he is caught stealing, the chances for victory drop to 42.0%. Should he steal? It depends on his chances of success.

Let “p” represent the probability of stealing 2nd base and 1 – p represent the probability of getting thrown out. The expected value for the Win Expectancy can be expressed as follows.

0.657 p + 0.420 (1 – p)

This simplifies to 0.237 p + 0.420. Stealing makes sense if the expected value of Win Expectancy increases from its original state of 56.7%.

0.237 p + 0.420 > 0.567

p > 0.620

Win Expectancy increases if the player has a greater than 62% chance of successfully stealing 2nd base. For what it’s worth, the total success rate in 2008 was 73%, but you have to consider that most often stolen bases are attempted by the top base runners and not your average runner. (A look at run expectancy suggests that the percentage needs to be above 75% to be beneficial in general. Perhaps I’ll address that in the future.)

Should the Next Batter Try to Sacrifice the Runner to 2nd Base?

There are 3 likely outcomes when a batter attempts a sacrifice bunt: runner safe/batter out (61.7% of the time during the 2003 season), runner out/batter safe (23.5%), and runner safe/batter safe (14.8%).

Here are the Win Expectancy values for these situations: runner safe/batter out (55.6%), runner out/batter safe (48.3%), and runner safe/batter safe (66.5%).

Let’s look at the expected value of Win Expectancy if the batter attempts a sacrifice bunt.

0.617 (0.556) + 0.235 (0.483) + 0.148 (0.665) = 0.555

So, the sacrifice bunt lowers the Win Expectancy from 56.7% to 55.5%, telling us that the sacrifice bunt hurts your chances of winning in this scenario.

Summary

Of course, the manager’s decision depends on the runner (is he an above average base stealer?) and the man at bat (is he a skilled bunter?). If you estimate the runner’s chances of a successful steal to be above 62%, the move increases your chance for victory. The sacrifice bunt hurts your chances for victory. I guess that’s why Dave Roberts stole 2nd in the legendary 4th game of the 2004 ALCS instead of having Bill Mueller attempt the sacrifice bunt. It was just a simple matter of mathematics!

-George

I am a math instructor at College of the Sequoias in Visalia, CA. You can reach me through the contact page on my website – http://georgewoodbury.com.