# combinatorial – expected benefit of my simple board game

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How to play:

Use 1 host and at least 1 player.

Each player has to throw six-sided dice to go to the goal.

If the player is in the 34th cell and throws 2 or more, he can go to the goal a as he has cast 1.

If the player reaches the goal in 9 pitches or less, the host must pay that player \$ 1 for 1 by throwing less than 10.

For example, if the player reaches the goal in 7 pitches, the host must pay \$ 3 to that player.

If the player reaches the goal in 11 pitches or more, that player must pay to receive \$ 1 for 1 throwing more than 10.

For example, if the player reaches the goal in 12 pitches, that player has to pay \$ 2 to the host.

If the player reaches the goal in 10 pitches, nobody has to pay.

Each game will end only if the player reaches the goal.

The player can not pay \$ 1 and start a new game if he can not reach the goal in the 11th pitch.

What is the expected benefit of host per player for each game?

From what I know of this game, the expected value to roll a six-sided die is 3.5.
The expected value of the distance in 10 launches is 35 cells, but the target is at a distance of 36 cells, so the expected benefit of the host is positive.
But I have no idea how to calculate.