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Steven and I have a biased coin. The coin has a 90% chance to show heads, and a 10% chance to show tails. We flip the coin in the same way 4 times. Steven picks heads for the first two rounds and I pick tails for the first two rounds. Then I choose heads for the last two rounds and Steven chooses tails for the last two rounds.

In order to win, one of us must succeed 3 times. Is one of us more likely to win?

My calculation shows that we each have a 0.1557 chance of winning. My friend is arguing that I am wrong based on absorbing markov chains.

Here's what I've done:

https://www.scribd.com/document/338158368/Swap-Side-Balance

Am I crazy or is this really obvious that it doesn't matter which side you start on?

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  • $\begingroup$ Which player does your friend claim will win more often and why? $\endgroup$ – whuber Oct 19 '17 at 14:44
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Before looking at the precise numbers, this looks essentially symmetric if you toss four times and so each player has the same probability of winning. Stopping after somebody has three should not change this

Now looking at the probabilities, Steven wins if:

  • HHH probability $0.9^3=0.081$
  • HHTH probability $0.9^3\times 0.1 = 0.0729$
  • HTHH probability $0.9 \times 0.1^3=0.0009$
  • THHH probability $0.9 \times 0.1^3=0.0009$

which is $0.1557$ as you say

You win if

  • TTT probability $0.9 \times 0.1^2=0.009$
  • TTHT probability $0.9 \times 0.1^3=0.0009$
  • THTT probability $0.9^3\times 0.1 = 0.0729$
  • HTTT probability $0.9^3\times 0.1 = 0.0729$

which is also $0.1557$

You can calculate the probability of a two-two tie as $0.9^4+4\times 0.9^2\times 0.1^2+0.1^4=0.6886$, making the total probability $1$

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Check out the 'Monty Hall' problem ( https://betterexplained.com/articles/understanding-the-monty-hall-problem/ )

If you assume you know, in advance, what will be the odds then you will get problematic logic. In this case, you assume both of you know this coin is biased, and what the odds are. Thus, the 'betting strategy' makes no difference: either of you can choose the first two (or last two) tosses, and things will be equal.

Things come out very differently if either (1) only one of you knows the facts, or (2) neither of you knows the facts.

People have written long papers on the Monty Hall problem. In the end - all math aside - it comes down to the fact he knows something you do not. A great deal of the field of Information Theory is about dealing with the balance of the known and unknown.

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