Thoughts on Coin Flipping

It’s Super Bowl weekend, otherwise known here as the weekend I lose 5 bucks to my friend Mattbo. Matt and I have a standing wager every year on the Super Bowl coin flip, and I seem to have an uncanny, almost scary, ability to lose money on the flip. I also lose money to Matt on Thanksgiving annually when my public school alma mater is routinely thrashed by Matt’s catholic school, but that’s a story for another time.

Coin flipping seems vanilla enough. It’s 50-50 probabilities make it seemingly uninteresting to study. But beneath the surface are lots of puzzling nuggets worth sharing with your students.

The NFC has won the opening coin toss in the Super Bowl for 14 consecutive years.  Go back and read that again slowly for maximum wow factor.  This is the sort of fascinating result which seems borderline impossible to many and brings on rumors of fixes and trends, but just how impressed should we be by this historical result?  Try simulating 45 coin flips (to represent the 45 Super Bowls) using a trusty graphing calculator.  What “runs” do we see?  Does having 14 in-a-row seem so implausible after simulation?  A number of sites (mosty gambling sites) have examined this piece of Super Bowl history, where some attach a probability of  1/16384 (2^14), or .00006 to this event.  But what exactly is this the probability OF?  In this case, it is the probability, starting with a given toss, that the NFC will win the next 14 in a row.  But it is also the probability that the AFC will win the next 14.  Or that the next 14 will be heads.  Or tails.  The blog The Book of Odds provides more information about the coin toss, specifically how it relates to winning the big game:

The odds a team that wins the coin toss will win the Super Bowl are 1 in 2.15(47%).

The odds a team that loses the coin toss will win the Super Bowl are 1 in 1.87 (53%).

The odds a team that calls the coin toss will win the Super Bowl are 1 in 1.79 (56%).

UPDATE:  In 2007, NPR ran a short piece during “All Things Considered” about the coin-flipping run, which was then in year 10.  Finally found it here.   It’s a quick 4-minutes and great to share with classes.

UPDATE #2:  The AFC just broke its dry spell.  Thanks to NBC for the nice stat line:

Exploring runs in coin tossing through simulation allows us to make sense of unusual phenomena.  On the TI-84, the randint feature allows for quick simulations (for example, the command RandInt (1,2,100) will produce a “random” string of 100 1’s and 2’s).  Deborah Nolan, a professor and author from UC Berkley, has developed an activity which challenges students to act randomly.  A class is split in half and given a blackboard for recording coin flipping results, and the professor leaves the room.  One group is charged with flipping a coin 100 times, and recording their results accurately.  The second group is given the task of fabricating a list of 100 coin flip results.  After both are finished, the professor returns and is able to quickly identify the falsifies list.  Too few runs give the fabricators away.

Does the manner in which a coin is tossed make the outcome more or less predictable?  Engineers at Harvard built a mechanical flipper to examine the relationship between a coin’s initial and final positions.  The assertion that much of the randomness in coin flipping  is the result of “sloppy humans” is tasty; we humans have trouble being random when needed.  Along the same lines of innovations in coin tossing, the 2009 and 2010 Liberty Bowl football games used something called the eCoin Toss to make the toss more accessible to the crowd.

Finally, if you are into old-school, bare-knuckles, coin flipping, you can mention these scientists, who each took coin flipping to the extreme:

Comte de Bufton:  (4,040 tosses, 2,048 heads)

Karl Pearson (24,000 tosses, 12,012 heads)

John Kerrich (10,000 tosses, 5,076 heads)