7. A Remarkable Coin

In this MathLab, the entire class works together as one large team. The students are simply asked to produce a coin that, after say 6 consecutive flips, came up heads on every single one!

Probabilistically, we need at least 2^6 or 64 coins. To be safe, we collect some 80 to 100 coins. Thus, each one of 20 students can flip 4 or 5 coins. After each flip, only those that came up heads are kept for the next flip. Sure enough (well, almost sure enough!) there will be at least one coin of the original 80 that came up heads six consecutive times!

Of course, many variations on this theme are obvious, such as we could also keep track of the coins that kept coming up tails; we can ask for 8 (or any number) in a row, etc., but we’d need more coins to start with. Over time, it would be cool for schools to eventually exhibit a coin in a display case with a caption something to the effect that, “This coin came up heads for 18 consecutive flips on Thursday, May 10, 2012.”

Students will learn something profound about probability theory. Perhaps the main point is that, while it is difficult to make a particular coin flip heads 6 to 8 times in a row, it is actually pretty easy to produce some coin that does so. By the way, students should be convinced that any winning coin is indeed a “fair coin” by recording its outcomes for many more flips after its remarkable run. The result is likely to be very unremarkable!

We see again that all three MathLab criteria are present in this activity:

Q. The object is to produce a coin that flipped heads 6 consecutive times.

R. To produce such a coin requires that we begin with a lot of coins. Probabilistically, if we begin with 64 coins, roughly 32 will flip heads. Of these, roughly 16 will flip heads after the second flip. Then about 8 of 16, then about 4 of 8, then about 2 of 4, then about 1 of 2. … But, to be safe, we should begin with many more than 64.

V. This activity is verified when, indeed, at least one coin fulfills the request.