# The Coin Toss Puzzle and the Simplest Possible Solution

Recently, I wrote an article on why the Monty Hall problem has perplexed so many brilliant minds where I showed that it was a corner case between 1 open and 1 closed door, while the intuitive but wrong answer is close to the probability curve of 1 open door.

Now a coin toss puzzle has appeared on Twitter [1] that has gone viral as it goes against our common intuition of probability and random sequences (such as a series of coin tosses). The puzzle goes as follows.

## The Problem

Flip a fair coin 100 times—it gives a sequence of heads (H) and tails (T). For each HH in the sequence of flips, Alice gets a point; for each HT, Bob does, e.g., for the sequence THHHT Alice gets 2 points and Bob gets 1 point. Who is most likely to win?

• Option A: Alice
• Option B: Bob
• Option C: Equally likely

I invite you to pause here for a moment and come up with your own solution.
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• Since coin tosses are the very foundation of our understanding of probability theory that can be best understood by equally likely outcomes, most people tend to think that Option C is the correct answer.
• Since Alice gains two points in 3 heads appearing together (HHH) and any extension thereof, as opposed to Bob who does not have this luxury, the second most chosen answer was Option A.
• In reality, the correct answer is Option B. Bob wins most of the times.

Let us find out the reason from an intuitive viewpoint.

## Intuitive Reasoning

For any sequence starting with HH, the next coin flip has one of the following two outcomes.

• HH.H: Alice wins with the score 2-0 (since in 100 tosses, Alice wins 99-0, 98-1, and so on, this margin of victory can easily be confused with the number of branches in which Alice wins).
• HH.T: It is a tie with the score 1-1 (HH for Alice and HT for Bob).

So we have 1 win for Alice and 1 draw between the two.

On the other hand, for any sequence starting with HT, the next coin flip has one of the following two outcomes.

• HT.H: Bob wins with the score 0-1.
• HT.T: Bob wins with the score 0-1.

So we have 2 wins for Bob in such a case, as opposed to Alice’s case above. The sequences starting with TH and TT do not change any result. Clearly, Bob wins in most branches of all possible sequences.

### References

[1] https://x.com/littmath/status/1769044719034647001?s=46&t=41cpGRZavPniB0OfiIMsDA