What would be the probability of winning the following game?

Question

The game is played on a board with twenty-four squares and a starting square. The player begins by flipping a coin. If they get heads they roll a die and move their piece the indicated number of spaces. If they get tails then they do not move anywhere. This is considered one turn. A player has a total of 12 turns to get their mover to the twenty-fourth spot, or else they lose.What would be the probability of winning the following game? The game is played on a board with twenty-four squares and a starting square. The player begins by flipping a coin. If they get heads they roll a die and move their piece the indicated number of spaces. If they get tails then they do not move anywhere. This is considered one turn. A player has a total of 12 turns to get their mover to the twenty-fourth spot, or else they lose.

Answer 1

You are performing the same experiment 12 times. Let the random variable x_i represent the number of squares moved at each trial.

 

The x_i are independent identically distributed.

 

You want to calculate P(x_{1 +} x_{2 +} ... + x₁₂ > 23).

 

You can of course find the joint distribution (convolution) of the summation.

 

 

Answer 2

Justin,

 

I would suggest computing the probability that the player loses.

 

So, compute

 

P(losing) = P(No H in 12 turns)⋅P(losing | No H) + P(1 H)⋅P(losing | 1 H) + ⋅⋅⋅ + P(12 H)⋅P(losing | 12 H)

 

and then take the complementary probability.

 

Regards,

Hassan