Probability of winning at least two games in a row - - - Elementary Probability

checkittwice

Member
You can play against player A or player B in an all-skill game
(such as chess or checkers).

Suppose there are no ties/draws.

On average you beat player A 90% of the time in this game,
and on average you beat player B 10% in this game.

You will play three games in row, and each game will be
against one player at a time.

You will choose one of these scenarios:

1st game - - a game against player A
2nd game - - a game against player B
3rd game - - a game against player A

OR

1st game - - a game against player B
2nd game - - a game against player A
3rd game - - a game against player B

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Which scenario should you choose to have the greatest
chance of winning at least two games in a row? * *

** This is adapted from a problem presented by Martin Gardner.

CaptainBlack

Well-known member
You can play against player A or player B in an all-skill game
(such as chess or checkers).

Suppose there are no ties/draws.

On average you beat player A 90% of the time in this game,
and on average you beat player B 10% in this game.

You will play three games in row, and each game will be
against one player at a time.

You will choose one of these scenarios:

1st game - - a game against player A
2nd game - - a game against player B
3rd game - - a game against player A

OR

1st game - - a game against player B
2nd game - - a game against player A
3rd game - - a game against player B

-------------------------------------------------------------------------------------

Which scenario should you choose to have the greatest
chance of winning at least two games in a row? * *

** This is adapted from a problem presented by Martin Gardner.

The second.

Construct a contingency tree to investigate further.

In both such trees there is a branch where the first two games are won, this branch occurs with probability $$0.09$$ (the outcome of the third game does not effect the probability of winning two games in a row along this branch).

The other main branch involves loseing the first game and winning the remaining two. This occurs with probability $$0.1 \times 0.1 \times 0.9$$ in the first case and $$0.9 \times 0.9 \times 0.1$$ in the second.

CB