Question

At carnivals and entertainment parks, there is a popular game of horse-racing. Each player has a movable horse on the display whose number corresponds to his or her alley number. The game consists of rolling a ball up an alley, where it can land in one of 15 holes. The holes are colored according to the graphic in the margin. If your ball lands in a red hole, your horse advances one space. If it lands in yellow, your horse advances two spaces, and if it lands in green, your horse gallops ahead three spaces. After your ball goes in the hole, it returns to you for your next roll. The first horse that moves 12 spaces is the winner, and the jockey, the person rolling the ball, receives a stuffed animal as a prize. One customer is at the game, bragging that he always wins after eight rolls. You look over the game and estimate that you have a 40% chance of getting the ball in a red hole, a 30% chance of getting it in a yellow hole, and a 20% chance of getting it in a green hole. There is also a 10% chance that you miss the holes, and your ball returns to you without advancing the horse. Based on your estimates of the probabilities, is it worthwhile to play the game against the bragging customer? What is the chance you will win by the eighth roll? EXCEL SHEET formulas!

Answer 1

Solution

Strategy:

Compute the expected number of steps the horse would advance in 8 rolls [the number claimed by one customer]. If this expected number is 12 or more, play the game; otherwise, do not play.

Concept Base:

Expected number of times the ball would land in a hole of a particular color = 8 x probability the ball would land in a hole of that color

Expected number of steps the horse would advance in 8 rolls

= Σ(Expected number of times the ball would land in a hole of a particular color x number of steps the horse advances for that color) – summed over all possible colors, including no color (i.e., miss the hole)

Now to work out the solution,

Summary of Excel calculations is tabulated below:

Number of Rolls		8
Hole Color	Prob	E(N)	# of steps	E(Steps)
Red	0.4	3.2	1	3.2
Yellow	0.3	2.4	2	4.8
Green	0.2	1.6	3	4.8
Miss Hole	0.1	0.8	0	0
Total	1	8	6	12.8

Vide ‘strategy’ above, since 12.8 > 12, decision is to play. Answer

Excel details

Cell labels

Number of Rolls		g40
Hole Color	Prob	E(N)	# of steps	E(Steps)
Red	f42	g42	h42	i42
Yellow	f43	g43	h43	i43
Green	f44	g44	h44	i44
Miss Hole	f45	g45	h45	i45
Total	f46	g46	h46	i46

Excel commands:

G42 = (f42*$g$40); G42 = (g42*h42);

Drag g42 down to g45 and i42 to i45

F46 = SUM(f42:f45) and drag f46 to i46

DONE