Suppose that there is a thin, circular ring of wire with circumference of 10 meters, and that we have some ants with the following properties,

1) All the ants when placed on the wire will travel at a fixed, constant speed of 1 meter/minute in the direction that they are heading.

2) When any two ants collide on the wire, each ant will instantaneously turn around and travel in the opposite direction at the same speed.

Suppose that we simultaneously place 14 ants at random locations on the ring. The orientation in which each ant is headed (clockwise or counter-clockwise) is determined by a fair, random coin flip. 13 of the ants are colored black, and 1 of the ants is colored red. We let the ants run around the ring as specified by the conditions above. After precisely 10 minutes, what is the exact probability that the red ant will end up at the spot it started in the beginning of the game?

Extra bonus:

The red ant would always return to the original position if and only if (1) the original orientation of all ants is the same, or (2) the number of ants with an initial clockwise orientation equals the number of ants with an initial counter-clockwise orientation.

## Question

## bushindo 14

Suppose that there is a thin, circular ring of wire with circumference of 10 meters, and that we have some ants with the following properties,

1) All the ants when placed on the wire will travel at a fixed, constant speed of 1 meter/minute in the direction that they are heading.

2) When any two ants collide on the wire, each ant will instantaneously turn around and travel in the opposite direction at the same speed.

Suppose that we simultaneously place 14 ants at random locations on the ring. The orientation in which each ant is headed (clockwise or counter-clockwise) is determined by a fair, random coin flip. 13 of the ants are colored black, and 1 of the ants is colored red. We let the ants run around the ring as specified by the conditions above. After precisely 10 minutes, what is the exact probability that the red ant will end up at the spot it started in the beginning of the game?

Extra bonus:

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