Categories
Statistics

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.

Powerball

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.

THE TABLE HAS BEEN SET, NOW FOR THE MAIN COURSE

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!

Categories
Statistics

The NFL Draft: Shopping for Bargains!

Last week, the NFL player draft took place over 3 days in New York City, and now the annual exercise of “grading” each team based on their draft haul commences.  It’s a fun debate, with grades often based more on feel or perceived value, rather than any real analysis.

There are many ways to evaluate draft results, but from a purely mathematical standpoint, I like to look at value.  Which teams  got the best “bargains”, and which teams went out on a limb?  If you had the 20th pick in the draft, did you get the 20th best player?  Or did you draft a lower-ranked player.

I took all of this year’s 254 players drafted in the NFL draft, and compared them to their draft ranking, according to CBS Sports.  The only real reason I have for using CBS as opposed to the many other draft rankings out there, is that it was easy to pull their data out into a spreadsheet.  From there, I computed the “value” of each pick.  If a team drafted a player above his rank, this is negative value.  If a team drafted a player after his rank, this is a positive value.  Some examples:

Geno Smith was drafted with the 39th pick, but was ranked 21st by CBS Sports, so his value was +18

Meanwhile, E.J. Manuel was drafted with the 16th pick, but was ranked 40th, for a value of -24.  

Some players represented great values for the teams which picked them:

Cornelius Washington, Chicago Bears (pick 188, ranked 82, +106)

Andre Ellington, Arizona Cardinals (187, 88, +99)

Jordan Poyer, Philadelphia Eagles (218, 119, +99)

While other players could be considered “reaches”:

B.J. Daniels, SF 49ers (pick 237, ranked 818, -581)

Jon Meeks, Buffalo Bills (143, 834, -691)

Ryan Seymour, Seattle Seahawks (220, “1000”, -780).  Ryan is the only drafted player who did not appear in CBS’s top 1000, so I just assigned him #1000.

There is a bit of un-fairness here, as many teams will use later picks on “projects”, players who have little expectation of making the team, but who seem to have a particular upside, so there was much volatility in the later round values.

From there, I simply added up the value scores for the players drafted by each team, and found an overall value score.  So, which teams earn the best grades?  Only 3 teams earned overall positive scores.  This is understandable, as it is much easier to earn negative scores than positives, especially in the later rounds.

THE TOP 3:

Minnesota Vikings (+187)

Chicago Bears (+51)

Philadelphia Eagles (+25)

THE BOTTOM 3:

Buffalo Bills (-836)

SF 49ers (-1097)

Seattle Seahawks (-1571)

For math class, have your students think of other ways to measure draft success.  Is the value measure here valid?  How can the method be adjusted?  How do some of the huge negative numbers in this data influence results?  Feel free to download and toy around with the data in my draft value tracker, and let me know what you come up with!

Categories
Statistics

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.