Tag Archives: probability

Class Opener – Day 6 – Foul Balls

In today’s opener, the Cleveland Indians provide the hook:

The article from ESPN.com provides the math:

The odds are one in a thousand just to catch one foul ball at any give game, according to ESPN Stats & Information. So what are the odds of one person catching four at a single game?

A cool one in a trillion, or simply a great day in Cleveland.

Pretty long odds. Almost suspiciously long.

Digging deeper, we can find out where ESPN came up with their trillion odds (linked from theblaze.com):


After students looked over this arguement, I asked them how many of them had caught foul balls at a baseball game before.  Where did they sit?

I was behind home plate.

I was near third base.

I was in the upper level near first base.

PNC parkFew people from the top sections, the outfield (where clearly a foul ball would not be an issue), and some other goofy sections ever have a chance at a foul ball.  This led to general agreement that some sections are “ideal” for catching foul balls, while others are not so great.  It appears that our Cleveland friend was probably sitting in one of the “hot” sections. The cool site IdealSeat provides heat maps for a number of MLB stadiums, showing you where to sit in order to optimize our chances of catching a baseball. Based on this evidence, is it safe to assume that the probability of catching a ball at a game is 1/1000? We agreed it was probably something lower, based on the lucky man’s seat location.

This is also a great time to talk about the multiplicationrule for independent events, where we agreed that the rule was (for the most part) applied correctly, though with some uneasiness.

I’ve reached the point where anytime probability or odds are quoted on the TV news or in a newspaper or magazine, I immediately am skeptical of the claim. I hope I transfer this desire to dig deeper to my classes.

Counting Principles and “The Price is Right”

I have a confession to make…..it’s really quite embarassing…

I’m a Price is Right nerd.

{sigh} wow, feels so good to get that off my chest.

priceSince I was a little kid, I loved watching the Price is Right.  I know all the games, many of the prices, and of course can name the “back in the day” models without batting an eye (Dian, Holly, Janice).  I even made the pilgramage to Television City a few years back to see Bob Barker in his last years of hosting.

Now, as a stat teacher, I have used a number of Price is Right games in the classroom as probability lessons.  I’ve given a number of talks using Plinko as the centerpiece of a lesson.  Almost all of Price’s games have some probability element.  Here are a few games you can discuss on your classes, starting with basic ideas, and moving up to more complex counting principles.


DoublePricesThis is the most simple probability game on the show.  The contestant is shown a prize, and two possible prices for the prize.  If the contestant guesses blindly, they then have a 50% of choosing the correct price, and winning the prize.  In all of these games, the pricing aspect is a “clue” to the player, which hopefully increases their chance of winning.  But in all of these examples, we will look at the games as random chance experiments.


This game is only slightly more  difficult than Double Prices.  In it, the contestant is shown three prizes, each with a price tag, one of which is an incorrect price.  If the contestant identifies the incorrect price, they win all 3 prizes.  Given random guessing, a contestant has a 1/3 chance of winning.


Here’s where we start looking at some more interesting counting methods.  In this game, a contestant can win a large prize and a smaller prize by correctly giving the price to the smaller prize.

SafeThe smaller prize has 3 distinct digits in its price, which the contestant is given.  They must place the digits in the correct order to find the price.  With 3 digits to place in order, we have 3! = 6 possible prices.  BUT, in this game, the price always ends in zero (they don’t tell you this, but it’s always true), which means this is essentially a 50-50 game.  For example, if the 3 given digits are 0, 9 and 5 – then there are only two possible prices, $950 or $590.


A contestant has a chance to win a prize with 4-digits in its price.  A “dummy” price is given, like $5447, and the contestant must determine if each digit in the actual price of the prize is higher or lower than the digit in the dummy price.

BonkersTo make their guesses, the contestant places markers either above or below each digit in the dummy price.  If they are correct with all 4 digits, they win the prize.  If they are wrong, they can go back and make changes.  A total of 30 seconds is given to make as many guesses as they can, running back and forth between the game board and the guessing buzzer.

Each digit has 2 outcomes, higher or lower.  Since there are 4 digits, there are 2 x 2 x 2 x 2 = 16 different outcomes.  With only 30 seconds to make guesses, this game often comes down to how well the contestant uses their time to maximize the number of guesses out of the 16.


In this game, a contestant can win 3 different prizes: one with 2 -digits in its price, one with 3-digits, and a car with a 5-digit price.

The contestant is first shown 3 digits, 2 of which make up the price of the first prize.  The goal is to use as few “chances” as possible to get the correct price, which allows the player to move on the the next prize.  With 3 digits to choose from, there are in theory 3 x 2 = 6 possibilities.  But like Safecrackers above, the price will always end in zero; so there are only 2 real choices.

For example: if the given digits are 0, 3 and 5, then the only real possibilities are $30 or $50.  Often, the frustration in this game is associated with contestants who don’t know that all the prices always end in zero…which causes me to yell at my television.

ChancesMoving onto the next prize, 4 digits are given, 3 of which make up the next price.  In theory, there would be 4 x 3 x 2 = 24 choices here.  But again, given that the price will always end in zero, there are only 3 x 2 = 6 viable choices.

The goal in this game is to economize your Chances, so that you have a good number left to play for the car.  5 digits are given, all of which must be used in the car price.  In theory, this gives 5 x 4 x 3 x 2 = 120 choices.  But there are 2 ideas at play here: the price will end in zero AND the price will always begin wth 1 or 2.  Depending on the assortment of digits given, this reduces the number of possible prices a player needs to assess.


DisksThis is one of the most difficult games on the show to win, and is often played for a luxury car.  The game is played with 8 wooden disks, which are placed in a bag and shaken.  5 of the disks have number on them; digits in the price of the car.  The other 3 disks have red “strikes” on them.  A disk is drawn, and if a number is drawn, the player must tell which position in the car’s price the number represents.  If they are correct, the disk is removed from circulation.  And if the player is able to complete the price of the car before drawing all of the strikes, they win the car.

This game is a bit tricky to analyze, because often numbers are drawn repeatedly, as the contestant tries to narrow down where digits go in the price.  For an in-class analysis, let’s assume that the player knows the price, and is trying to just draw the digits.

If you are assuming perfect play, then we could simply list all of the possible ways to arrange the numbers and strikes.  Let N indicate a number and X indicate a strike.  So you could have:

NNNXXNXN (Loss, since 3 X’s occur before all N’s are drawn)
NNXNNXXN (Loss)…..

The number of ways to play the game is then 8C3, which is 56.  Now, this may seem like a small number, but I am treating the numbers in the bag as similar objects, since we are assuming the contestant places them correctly.

BoardNow, I could go through and count the number of these 56 that produce wins, but I think it might be simpler than that.  The game really comes down to the last item on the list.  If the last item is an X, then you will have won the game.  If the last item is an N, then you have lost.  The chance that the last item is an X 3/8, so the probability of winning the game, assuming perfect play, is 3/8.

So, this game only has  37.5% chance of victory IF the contestant plays perfectly.  Add in that often the contestant often must struggle to position the digits, and you see why the game is so difficult to win.


LineUpThis game is played for a car, with 3 smaller prizes as well.  The 3 smaller prizes have prices with 3 -digits, 2 digits, and 3-digits.  The prices of these smaller prizes are used to fill in the middle 3-digits in the price of the car, as shown on the game board here.  This give 3 x 2 x 3 = 18 possible outcomes.  The nice part about this game is that the contestant is given a second chance, and is told how many digits they have correct after the first attempt.  If the player needs their second attempt, it would be interesting to analyze how many choices of the 18 remain, given that they have 0, 1 or 2 of the digits correct.

There are plenty of other games on the show which also have basic counting prinicple ideas worth exploring.  Some quick hits:

  • Balance Game – how many different ways can 2 bags be chosen from the given 3.
  • Dice Game – how likely is it to roll correct digits?  When should I choose higher or lower?
  • Golden Road – how difficult is it to advance to and win the big price at the end of the Golden Road?
  • Let Em Roll – how many different ways can the 5 special dice be rolled?
  • Make Your Move – how many different possibilities exist for moving tre sliders?
  • Race Game – how many ways can the price tags be placed?
  • Take Two – how many ways are there to choose 2 prizes from the given 4?

And some in-class ideas:

  • Let students choose a game to analyze.  Create a poster and share with the class.
  • Start each day of your probability unit with “a Game a Day”.  Start with the easy games, and move it to the more complex ones.
  • Have a contest where students design there own pricing games.

Thanks to my friends at Golden-Road.net for the fun pictures.

The Case of “Too Many Powerball Winners”

Here’s a favorite activity of mine from prob/stat class.  I love bringing in real stories of statistical improbability from the media to get kids thinking about real-world applications of probability, and reinforce the fact that theoretical probability represents a long-term ratio.  In the short term, funky stuff happens sometimes.  In an earlier post, I gave some examples from the Amazing Race and the casino world.  Today’s example comes from another gambling example: Powerball.

First, some understanding of the game is required.  In Powerball, players attempt to guess the numbers that wil be drawn from ping-pong ball machines.  Two different machines are used for the game.  In the first machine, there are 59 white balls, while a second machine holds red balls numbered 1-35.


Players select 5 numbers they believe will be drawn from the white-ball machine, and 1 number they believe will be drawn from the red-ball machine.  If you match all 6 numbers correctly, you win the grand prize, often in the 10’s of millions of dollars.  For more info on the game and how to play, the Powerball website provides lots of info, including a rather amusing FAQ area.  You can also use random.org to generate some draws, play the game with your class, and hopefully show them how difficult Powerball is to win, or even get 2 numbers correct.


Print out the first page of this file, which is an article from the Washington Post, with key information removed.  Don’t give out the second page – it contains the secret to this probability anomoly.  You can read the article on your own, but here is a summary of the article:

  • If a player matches all 5 white balls in a Powerball drawing, but not the red ball, they win a prize of $100,000.
  • In a given week, there are “usually” 4 or 5 such winners.
  • On a drawing in 2005, there were 110 winners.
  • The Powerball police investigated.

What caused so many winners?  Cheating?  Luck?  Pure chance?  Are the winning numbers “special” in any way?  In Pennsylvania’s Daily Number, for example, the state pays has paid out more than 5 times the amount wagered when the 7-7-7 combination is drawn.

The second page of the article gives away the surprising twist, after students think about the situation, and make some conjectures.  So what’s the twist?

Spoiler space….if you want to think so more…do it now….



More spoiler space…..



Ready….it’s fortune cookies!

Fortune Cookies!

A company in Queens, NY produces fortune cookies for restaurants, and chooses numbers to go on the fortunes.  They seem to use the same numbers in a batch, and these numbers found their way into the hands of hungry Chinese-food lovers, who played the numbers.  They just happened to hit!

Hope you enjoy this tale of statistical improbability!

Monopoly Math

The big Monopoly battle is coming near its end, and the iron and racecar are battling for Monopoly supremacy.

Monopoly Board

Both players own properties on the next block, and have some spaces they’d like to avoid.  For the car, here are the spaces he’d like to avoid.

Car Spots

And for the iron, there are a few spaces to avoid.

Monopoly - Iron 1

Since there some houses and hotels on some of the spaces, they are worth different amounts.  Below, here is how much each player will have to pay if they land on the “bad” spaces.

Car Board B

Monopoly - Iron 2

So, here’s the question:  which player is in “worse” shape?  Which player should be more worried about their upcoming turn?

Let this stew with your classes, and would enjoy hearing some class reflections.  The big reveal will come in a few days.

Probability Openers – Separating the Possible from the Plausible

A recent problem I reviewed from the Mathematics Assessment Project (MAP) caused me to refelct upon the coverage of probability in our classrooms.  The website provides sample tasks and assessment tools for schools and districts as they adjust curriculums to match the Common Core.  In the standards, probability begins to take center stage in grade 7:

Investigate chance processes and develop, use, and evaluate probability models.

Probability is treated like the ugly step-sister in many math courses: ignored, shoved to the side.  Look at standardized testing results, including AP Statistics, and you will find that probability standards often produce weak results.  The isolated fashion with which we treat probability is certainly not helping.  Let’s develop strategies to not only re-think probability, but to encourage communication of ideas and develop understanding.

I have taught probability at many levels: as an 8th grade teacher, as an AP Statistics teacher (and reader) and as the author of a Prob/Stat course delivered to 9th grade students.  This year, I taught my first college course in Statistics, where the problems with probability persist.  The picture below is from my college Stat 1 class, which you can also see on the great site Math Mistakes, by Mike Pershan.  Visit and provide your input:


Here are two activities I hope you can use in your classrooms to help fight the probability battle.


The Core Standards provide a framework for our students base knowledge of probability:

Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

Before we can start diving into formulas and fractions with probability, we need students to understand, and be able to express, whether an event is likely or not likely.  For simple events, like flipping a coin or drawing a card, this can be an easy discussion, but what happens when we start talking about complex events, like flipping multiple coins, tossing 2 dice, or choosing a random car from a parking lot?  Download my Probability Opener, which asks students to assess events and compare their probabilities.  Answers are provided.  The events start off innocently enough, but soon meander into more complex tasks, where students are asked to identify the more likely outcome:

You roll the dice and move your piece in the game of Monopoly…

A:  You end up on Boardwalk

B:  You end up in Jail

This is also a great activity if you have a classroom clicker system, or for Poll Everywhere if cell phones are allowed.


This activity is an adaptation of a Spinner Bingo problem from the Mathematics Assessment Project site I mentioned at the top of the post.  Visit the MAP site for not only the task, but a rubric and samples of student work.

The problem presents a scenario where students are asked to assess a spinner game, and bingo cards created for the game.  Read the files given on your own, but here is a quick summary:

  • 3×3 bingo cards are filled out, using non-repeating numbers from 1-16.
  • Two spinners with 8 equal spaces numbered 1-8, are spun, and the sum computed.
  • Players mark off their bingo cards if the sum appears on their card.

This is a nice scenario, so let’s adapt it and use it as our unit opener.  Having students play a game, and develop and justify a strategy, is a great way to get started.  Here is a lesson guide I have written to help you get started.  Also, the site Unpractical Math provides a virtual spinner applet you can use.  It’s often easier to just dive in and play, so here is a brief video demo of the game and how you can play it in your classroom:

Please let me know if you use either of these activities, and would appreciate your feedback. Thanks.

Ring in the New Year with Fun Classroom Lessons!

Now is a good time to reflect upon the past year, and think about all of the professional growth I have made through the people whose ideas I have shared and experienced through the twitter-sphere and blog-o-sphere (are these actual words?), and to send thanks from all of the new math friends I have made.  I took a look back at all of my posts from the previous year, and here are 5 great activities you can use tomorrow is your classroom.  Share them, adapt them, expand upon them…it’s all good.  Just pay it forward and share your best works, or leave a comment /contact me and let me know if you use them!  Enjoy.

Conic Sections Drawing Project – this was the most popular post of the year.  For algebra 2 or pre-calc, this project just got better with the Desmos online calculator, which is my favorite new tool of the past year.

Tapping Into the Addition of Bubble Wrap – bubble wrap, iPads, and slope meet for a fun exploration.   Look at rate of change through student-produced data.

Tall Tales for Probability – Featuring the poker chip drawing game, and examples from the Amazing Race and craps.  Probability should be fun.  Make it so!

Let’s Play Plinko! – I have used Plinko as an introduction for binomial distributions for years, but in this presentation from last summer’s Siemens STEM Academy, tech tools like PollEverywhere and Google Drive are used to increase interaction.

Composite Functions and ESP – Use this activity with middle-schools and see if they can develop the pattern.  For high school, have students write and justify their own ESP puzzles.  Also features Doceri, another favorite new tool of mine, for iPad.

Tall Tales for Probability Class

Probability is fun.

Nothing drives me nuttier than boring probability units.  Endless worksheets filled with tales of balls in urns and cards being drawn from decks.  Nothing screams for fun and games more than a probability unit.  Here’s some ideas for you to try:


This is a good opener for teaching dependent events.  Take a brown paper lunch bag, and place inside 20 colored cubes.  In the past, I have used 10 white, 6 red and 4 blue.  Do NOT let the students know what chips are in the bag.  Shake the bag, and travel from student to student, having each student draw 3 cubes from the bag (without replacement, or all 3 at once, doesn’t matter).  If they are able to draw exactly 2 whites and a red from the bag, they receive a Jolly Rancher reward.


After every student has had a chance to play, I add an additional challenge to the game.  Letting the class know that there are exactly 20 cubes in the bag, can they predict the distribution of colors in the bag, using the information gathered from the pulls they observed as evidence?  Students who can guess the exact distribution also win the coveted Jolly Ranchers.

Next, dump out the cubes to show the true distribution.  Was the 2 white, 1 red game “winnable”?  Was it possible to win?  Was it plausible that one would win?  I usually try to rig the color distribution so that one can win about 25% of the time.  The 10 white, 6 red, 4 blue configuration will provide a 23.6% probability of winning.


Always remind your students that probability is a long-term ratio.  The 1/6 probability of rolling a five on a die does not mean that every 6th roll will be a five.  In the short term, strange things can happen.

The CBS show “The Amazing Race” provided a great probability teaching moment in its 6th season.  In a “Pit Stop”, teams must complete a challenge in order to earn a clue and move on in the race.  Here’s a summary of the challenge:

In a field are 270 large hay bales.  20 of the bales contain clues.  You must continue to roll out hay bales until you locate one with a clue, at which time you can move on.

Think about these openers:

  • What is the probability you select a bale with a clue on your first attempt?
  • What is the probability you select a bale with a clue on your second attempt?  Third attempt?
  • What is the probability it takes you 5 or more bales to locate a clue?
  • What is the probability you roll out 20 bales, and do not find a clue?

The clip from this show appears below.  You’ll love the reactions of your students!

One last example tale comes from the world of gambling.  While I try to stay away from too much gambling talk in classes, this die-rolling activity leads to a neat backstory.  Have your students roll pairs of dice, and record the number of rolls needed to roll a sum of 7.  After a 7 is rolled, start a new count.  Plotting the results in a class dotplot gives a nice example of a skewed-right distribution, which is often a new shape for our students.

In class, what was the most number of rolls needed?  I have had students get into the 20’s or 30’s.  Is it possible that it could take 50 rolls to get a sum of 7?  Is it plausible?  Note the subtle language lesson happening.

Now, on to craps.  Craps is a fairly complex game, which boils down to this for our example:  once a player rolls, they continue winning until a sum of 7 is rolled, at which point the round ends.

Meet Pat DeMauro:

Craps winner

Pat visited the Borgata casino in Atlantic City in May, 2009 and set the world record for a game of caps, by tossing the dice 154 times without rolling a 7!

Pat’s winning streak was featured in Time Magazine, and features some of the details of the record run.

Probability is exciting!  Make it so!