Carnival Game Probability: Calculating Expected Value

[brickwalsh]

1.

Homework Statement
In a carnival game, a rectangular board is set up with red circles 1 inch in diameter painted on it in a grid. The centers of the circles are 3 inches apart and the centers of the outer rows of circles are 1.5 inches from the closest edge of the board. A player tosses a dime on the board. If the dime lies entirely within a red circle, the player wins a prize worth $5, but if any part of the dime lies outside the circle the player loses. Suppose the probability the dime falls off the board is .1 and the diameter of a dime is 11/16 inches. What is the expected value of the game to a person playing the game? Assume the player does not get his dime back even if he wins a prize.

The Attempt at a Solution

I'm really struggling with this problem. I think I need to do $5*P(dime completely lands in a red circle)-.1*P(dime misses a circle or falls off the board), but have no idea where to go from there.

 

[Ray Vickson]

1.

Homework Statement
In a carnival game, a rectangular board is set up with red circles 1 inch in diameter painted on it in a grid. The centers of the circles are 3 inches apart and the centers of the outer rows of circles are 1.5 inches from the closest edge of the board. A player tosses a dime on the board. If the dime lies entirely within a red circle, the player wins a prize worth $5, but if any part of the dime lies outside the circle the player loses. Suppose the probability the dime falls off the board is .1 and the diameter of a dime is 11/16 inches. What is the expected value of the game to a person playing the game? Assume the player does not get his dime back even if he wins a prize.

The Attempt at a Solution

I'm really struggling with this problem. I think I need to do $5*P(dime completely lands in a red circle)-.1*P(dime misses a circle or falls off the board), but have no idea where to go from there.

Since the dime is gone under any outcome, just forget it and subtract it at the end of your computation. So, there are only two relevant events and outcomes: W = win, with gain $5 and L = lose, with gain $0.

You need to compute the expected gain, so you need to figure our P(W). Where must the center of the dime fall in order that W occur? Assuming that the center of the dime falls uniformly over the board (given that it falls on the board at all), how can you get P(W)?

 

[brickwalsh]

Since the dime is gone under any outcome, just forget it and subtract it at the end of your computation. So, there are only two relevant events and outcomes: W = win, with gain $5 and L = lose, with gain $0.

You need to compute the expected gain, so you need to figure our P(W). Where must the center of the dime fall in order that W occur? Assuming that the center of the dime falls uniformly over the board (given that it falls on the board at all), how can you get P(W)?

I think the center of the dime has to land within 5/16 inches of the center of the circle. What I can't figure out is how to find the probability when you don't know how many circles are the board. Also, how do you know it's uniformly distributed?

 

[Ray Vickson]

I think the center of the dime has to land within 5/16 inches of the center of the circle. What I can't figure out is how to find the probability when you don't know how many circles are the board. Also, how do you know it's uniformly distributed?

I don't know that the dime falls uniformly; I said "assuming...". If you don't like the uniformity assumption---and, truly I do not like it myself---you need to make some other assumptions about the form of the distribution. (I have seen similar models discussed in textbooks, using a bivariate normal distribution for the position of the center of the dime. Of course, you would need a mean vector and a variance-covariance matrix to specify the distribution, and the answer would be obtainable only through multivariate numerical integration.)

That aside, the question is whether the value you get for P(W) will depend on the size of the board. Will it be different for a very small board having, say, 4 circles, vs. a very large board having millions of circles? Look carefully at the problem before deciding.

 

[brickwalsh]

okay Ray thanks for your help!