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Alice and Bob play a game. Bob can play $\$1$ to flip coin. If he gets three consecutive heads, like HHH, then he gets $\$12$. Is this a good game for Bob?

I think the answer is yes but I can see why it can also be no. In this thread Expected number of tosses to get 3 consecutive Heads it is shown that the expected number of coin flips to get 3 consecutive heads is $14$. So that makes me think that it's unfair for Bob. But then I tried doing a markov chain calculation by writing the transition matrix

$$\begin{bmatrix}0.5 & 0.5 & 0 & 0 \\ 0.5 & 0 & 0.5 & 0 \\ 0.5 & 0 & 0 & 0.5 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$

First state is where we start or where we go when we get a tails (no heads in the sequence). Second state is with one heads, third state is with two heads, and fourth state is with 3 heads.

I calculated the matrix to the power of $12$ and I looked at the top-right corner entry, and it was $0.58$. So this makes me think yes, it is a good game for Bob because $0.58 > 0.5$ which means you expect to get into HHH state with 12 or less coin flips.

What do you think? Which interpretation is right?

  • Your first analysis is correct. It is not a good game for Bob. I'm not well versed in Markov chains, so I can't comment about your second approach. – Rushabh Mehta Oct 08 '19 at 21:22

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The answer depends crucially upon what you mean by "he gets \$12".

If you mean that he has paid \$3 but then wins \$12, it is not a good game for him.

However, if he is effectively paying \$3 but winning \$15 then this is a good game for him because $15>14$.

  • Yes he pays $3$ and wins $12$. Why is the markov chain analysis wrong? –  Oct 08 '19 at 22:33
  • The 0.58 means his probability of having just won $12 is 0.58. However he will have paid $12 to have this number of tosses. This is not a good reason to play the game! –  Oct 08 '19 at 22:44