Coin Flips

The following article is an excerpt from the "Ask Marilyn" column from the PARADE section of the Chicago Tribune newspaper. Marilyn vos Savant is currently listed in the "Guinness Book of Records" Hall of Fame for "Highest IQ."

Reader's Question:

I have just tossed a coin 10 times, and I ask you to guess which of the three sequences below was the result (H=Heads and T=Tails). One (and only one) of the sequences is genuine:

(1) HHHHHHHHHH
(2) HHTTHHTHHH
(3) TTTTTTTTTT

Experts say that most people will choose #2 but that choice is not rational. According to probability theory, the chances of any of the 3 occurring are equal. Although I agree that the chances are equal, why is it irrational to choose the sequence with the most typical pattern in a genuine toss?

Marilyn's Answer:

I know I'm violating a cherished notion among probability experts, but I don't believe choosing #2 is irrational. Though the chances of the above three specific sequences occurring randomly are equal, they weren't randomly included in the problem, so they can't be given equal weight. Instead, only one of the sequences was randomly generated; the other two were chosen. As we know this fact, it's reasonable for us to choose #2 as the most likely genuine result. That's because far more randomly generated sequences will be a mix than all heads or all tails. (Note: If you ask before the coin toss which of the 3 specific sequences will be the most likely result, I'd agree with the original analysis: The chances are equal. But that's a different question.)

This is easy to test. Toss a coin 10 times and write down the result. Add two other choices: all heads and all tails. Ask your friends which sequence is genuine. Then toss a coin again and repeat the process. After a few dozen of these trails, the following will be obvious: (1) The genuine sequence will almost always be mixed; (2) everyone who guesses the mixed sequence will be correct; and (3) you'll have fewer friends than before.