At a County Fair, a dart game is set up so that the “bulls eye” is a circle of r

Question

At a County Fair, a dart game is set up so that the “bulls eye” is a circle of radius 2 cm, surrounded by two circles with the same center and radii of 4 and 9 cm respectively. To play one game you pay $1.50 and get three darts to throw at the board. If a dart does not land somewhere inside the outer circle, you get to try again. The payoffs for hitting the bulls eye and the surrounding circles are $1, 50 cents and 20 cents respectively. We will assume that all players throw the dart at random, so that it hits in some random spot inside the outermost circle. a. What is the probability of a player hitting the bull’s eye? What is the probability of hitting the inner ring? What is the probability of hitting the outer ring? b. From the probabilities calculated above, what is the expected payoff per game for a random player? Note: A game is three throws. c. If 1000 people play in this way (randomly throwing 3 darts each), what is the expected payoff to the game master?

Explanation / Answer

(A) Probability that a player hitting the bull's eye?

Pr (Hitting bull's eye) = Area of the bull's eye hit/ total area of dart game = * 22 / * 92 = 4/ 81

Pr(Hitting the inner ring) = Area of the inner ring/ Are of the dart game = * (42 - 22 ) / * 92 = 12/81 = 4/27

Pr(Hitting the outer ring) = Area of the outer ring/ Are of the dart game = * (92 - 42 ) / * 92 = 65/81

(b) Expcected Payoff for a random player for a random game =

FOr on throw expected payoff = 4/81* 1 + 12/81 * 0.50 + 65/81 * 0.20 = $ 23/81 = $ 0.284

For an game of three throws expected payoff = 23/81 * 3 = $ 0.852

(c) If 1000 people play in this way

Expected payoff to the game master = 1000 * (1.50 - 0.852) = $ 648

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