# Category Archives: Puzzles/Games

## THE “MEN & HATS” PROBABILITY PARADOX

Imagine that each of N=6 men has a hat. Also imagine that these hats are identical except that each man’s name is written inside his hat. Finally, imagine that the 6 hats are taken up and then later, because they look alike, randomly returned to the men.

As the 6 hats are returned to the 6 men, there’s a chance that no man will receive his own hat.  The chance of this happening is a tad greater than 1 in 3. To be more precise, the probability (to 3 decimal places) of all 6 hats going to the wrong individuals is .368.

Now, let’s add a new wrinkle to this  imaginary situation. Suppose the number of men (each with a hat) is greater than 6. What if there are 7 men? Or 8? Or more? As N increases, what happens to the probability that no hat will be returned to its proper owner? Some people guess that this probability goes up as N increases. Others guess that this probability goes down.

Both thoughts are wrong.

That’s because the likelihood of no correct “match” is virtually the same for any N > 5, whether N = 6 or N = 600 or N = 600,000!

The actual probability (p) of having no hat returned to its proper owner is given by this formula:

p  =  1/(2!)  –  1/(3!)  +  1/(4!)  –  1/(5!)  +  . . .

where there are N-1 terms on the right side of the equation. With the symbol “!” standing for “factorial,” we could rewrite the above formula as

p  =  1/2  –  1/6  +  1/24  –  1/120  +  . . .

As either of the above formulas shows, additional terms on the right side of the equation have a smaller and smaller impact on the value of p. Moreover, the drop-off of this impact is sharp, not gradual. This fact is made clear by the following chart showing the value of p, to 6 decimal places, for the case where N = 2, 3, 4, … , 10.

N = 2     p = .500000

N = 3     p = .333333

N = 4     p = .375000

N = 5     p = .366666

N = 6     p = .368054

N = 7     p = .367857

N = 8     p = .367882

N = 9     p = .367879

N = 10     p = .367879

It should be noted that this puzzle question is sometimes referred to as “Montmort’s Problem.” Montmort was a Frenchman who studied the probability behind a game called “Treize.” (Treize is the French word for 13.) In its original form, the puzzle question dealt with a jar containing identical balls numbered 1, 2, 3, … , 13. If balls are randomly pulled out of the jar, one at a time, the puzzle question was stated like this: “What’s the probability that the 1st ball taken from the jar will not be the ball numbered 1, that the 2nd ball will not be the ball numbered 2, and so on, with the end result being that no number on any ball matches the order in which the ball is removed from the jar?”

Filed under Mini-Lessons, Misconceptions, Puzzles/Games

## YOU LIKELY KNOW THIS PERSON’S NAME . . . BUT CAN YOU GUESS IT?

This important historical figure:

a) … coined the term “standard deviation”;

b) … had his book, “The Grammar of Science,” selected by 23-year-old Albert Einstein as the first item to be studied in Einstein’s elite “book club”;

c) … co-founded (1885) the “Men & Women’s Club” to discuss gender equality;

d) … has his main stats creation summarized by his last name’s middle letter.

FOR THE ANSWER: go to www.rutherfordjournal.org/article010107.html

## A Normal Curve Puzzle

First, think of a normal (Gaussian) curve. Don’t ponder its formula, origin, or application; instead, just picture it. To be more specific, create a mental image of the famous bell-shaped curve resting on a straight, horizontal “baseline” that’s labeled with z-scores.

In the picture now in your mind, the z-score on the baseline directly under the tallest part of the curve should equal zero, with that value corresponding to the mean. Other z-scores on the baseline, of course, simply indicate the lateral distance from the mean as measured in standard deviation units. The z-scores to the right of zero are positive numbers, corresponding to scores that are above the mean. Conversely, z-scores to the left of the mean are negative.

Clearly, the curved line above the baseline does not have the same degree of “curvature” at all points from far left to far right. When the curve is 10 standard deviations the right (or left) of the mean, the curvature is quite small. When this far out from the mean, the curve is nearly flat. Closer in, the curve has more curvature.

OK. Now that you have access to a mental image of the normal curve sitting on a baseline of z-scores, here comes the puzzle question:

At what z-score value(s) is the normal curve curved the most?

(If you come up with an answer to this little puzzle and would like to see if it’s correct, send your solution in an email message to Sky Huck at shuck@utk.edu