Puzzle of the Week for 16 November 1998: Solution

Although your friend does give you a penny that behaves exactly as described, it happens to be mixed in with about 2000 other pennies. She tells you to toss all of them, and about half land tails up. She takes these away and tells you to toss the remaining pennies (about 1000 of them). Again, about half of these land heads up.... Can you see what happens next?

After repeating this nine times, you have (probably) about 4 pennies that have landed heads up nine times in a row. You toss these and (probably) 1, 2, or 3 of them land heads up a tenth time. (If this doesn't work, just start over with 2000 pennies again!)

So you now have a most unusual penny (or maybe more than one) that has landed heads up ten times in a row. Has this procedure selected for pennies that are likely to land heads up? What do you think will happen if you toss this unusual penny ten more times?

If there really are any lopsided pennies that favor landing heads up (or, for that matter, any two-headed pennies), this procedure will find them. Unless your friend's penny collection is a lot more unusual than mine, however, the pennies you find will have no special tendency to fall heads up.

It may seem exceedingly unlikely that you can toss an ordinary penny 10 times and that it will land heads up 10 times in a row, and (in a single set of 10 tosses) this is indeed unlikely: the probability of this happening is 1 in 210, or a bit less than 0.1%.

In fact, the probability of seeing any specified sequence of heads and tails in 10 tosses is also 1 in 210. If we perform many, many sets of 10 tosses, we will see all of the possible sequences. On average, we will see any given sequence, such as 10 heads in a row, one time in every 1024 sets of 10 tosses. It's not surprising to find at least one penny among 2000 that behaves this way; in fact, it would be somewhat surprising not to find one.

Our tendency to seek patterns in random events can lead our intuitions astray. If we ask a thousand people to predict the results of ten coin tosses, we might find one who does so correctly each time. Does this person have ESP?