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Suppose you are tied $1-1$ with Bob. To win the game, you have win greater than or equal to $5$ rounds and win at least $2$ more rounds than your opponent. The probability of winning a round is $40\%$ and each round is independent. What is the probability of you winning the game?

This would be: $$\left[1-P(\text{winning less than 5 rounds}) \times P(\text{win at least $2$ more rounds than your opponent}\right]$$

$$ = \left[1-\sum_{i=0}^{3} \binom{3}{i} (0.4)^{i}(0.6)^{3-i} \right] \times \left[P(\text{win at least $2$ more rounds than your opponent}) \right] $$

What would be the second term? Would you just calculate 1-complement?

Damien
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  • The duplicate is the same question, merely with slight changes in the numbers. – whuber Jan 16 '19 at 13:29
  • @whuber: Is this the only solution? Seems very long and complicated. Would my approach using the complement work here? – Damien Jan 16 '19 at 13:45
  • It's not the only solution: several good approaches are presented in that thread. Of course they all have to produce the same answer! Since I don't understand what you mean by "1-complement," I can't give you an opinion on whether that will produce a correct answer. But you should easily be able to check, now that you have seen the right formula. – whuber Jan 16 '19 at 13:46
  • @whuber: The probability of winning is equal to the probability of winning at least 5 rounds multiplied by the probability of winning at least 2 more rounds than your opponent. The probability of winning at least 5 rounds is equal to (1-probability of winning less than 5 rounds). The probability of winning at least 2 more rounds than your opponent is equal to (1-probability of winning less than 2 more rounds than your opponent). Multiply these together to get desired probability. Does this answer make sense? – Damien Jan 16 '19 at 13:56
  • Almost: what justifies the final multiplication? That's tantamount to assuming the two events "winning at least 5 rounds" and "winning at least 2 more rounds than your opponent" are independent. Intuitively that's a doubtful proposition, so you ought to provide a demonstration if you think it's true. Regardless, you're in a great position to answer such questions yourself: write down the resulting formula and compare it to any of the formulas offered in the duplicate. – whuber Jan 16 '19 at 14:00

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