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 draw any color.
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: .