Who Gets The Ticket?

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Two players play a game of chance. The winner gets a free ticket to watch the Champions League semi-final between Bayern and Madrid (second leg).

They pull out a random guy from the street and ask him to think of a random positive integer between \(1\) and \(5\) (inclusive). The random guy keeps his choice secret. The players then take turns (starting with the first player) alternatively guessing the number.

The first player to guess the number correctly wins. The rules are, if one of the players guesses wrong, the random guy tells them whether the guess was smaller or larger than the number he thought.

If the probability that the first player wins is equal to \(\dfrac{a}{b}\) for some coprime positive integers \(a,b,\) find \(a+b.\)

Details and assumptions

  • The guesses of the players are completely random. However, since they have a basic idea of mathematics, the players don't guess a number already guessed before.

  • The random guy never lies.

  • Both players get to listen what the other player guessed, and what the random guy responded.


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