Hello, Yan!

I had to crank out a few scenarios to discern a pattern.A jar contains marbles of 7 different colors.

Each player draws a marble on his/her turn, and returns it to the jar.

A player wins the game by drawing the same color marble on two consecutive turns.

If there are three players, what is the probability that player 2 wins?

Give your answer as a fraction.

Call the players #1, #2, and #3.

On the first draw, each player can drawanycolor.

Suppose #2 wins in the second round.

. . #1 must not get his first color: .

. . Then #2 gets his color: .

Suppose #2 wins in the third round.

. . None of them get their number in the second round: .

. . #1 does not get his second color in the 3rd round: .

. . Then #2 gets his second color: .

Suppose #2 wins in the fourth round.

. . None of them get their first numbers in the second round: .

. . None of the get their second numbers in the third round: .

. . #1 does not get this 3rd color in the 4th round: .

. . Then #2 gets his 3rd color: .

See the pattern?

. . . . . . . .

The geometric series has: first term , common ratio

. . Its sum is: .

Therefore: .