Question

I am going to install a slot in the VIP lounge of my casino. A coin is inserted and a button is pressed. The screen reveals 3 colored circles (on the left, middle, and right side respectively). Each of the colored circles will be purple, red, blue, black, white independently with P[purple]=P[red]=P[white]=P[blue] = 0.5P[Black]. The payoff is as follows:

circles reward (coins)

3 white 5000

3 blue 1000

3 purple 100

3 red 6

2 red/1 non-red 4

1 red/2 non-red 2

all other results 0

a) Calculate the probability of each of the patterns occurring.

b) Calculate the player's expected gain.

Answer 1

Given

P[purple]=P[red]=P[white]=P[blue] = 0.5P[Black]

Let P[purple]=P[red]=P[white]=P[blue] = p

So P[Black] = 2p

Now, each circle can have one of these colors only. So the probability of each color should sum up to 1.

p+p+p+p+2p = 1 => p = 1/6

=> Let P[purple]=P[red]=P[white]=P[blue] = 1/6 and P[black] = 2/6 = 1/3

a) Since each circle will be colored independently so we can multiply probabilities of each color to get probaibility of those colors.

• P(3 whites) = P(white)^3 = p^3 = 1/216

• P(3 blue) = P(blue)^3 = p^3 = 1/216

• P(3 purple) = P(purple)^3 = p^3 = 1/216

• P(3 red) = P(red)^3 = p^3 = 1/216

• P(2 red/ 1 non red) = P(2 red)*P(1 other) = p^2 * (1-p) = 5/216

• P(1 red/ 2 non red) = P(1 red)*P(2 others) = p*(1-p)^2 = 25/216

• otherwise 0

b) Expected gain ?

We have 3 independent circles and five colors. So possible color patterns in three circles = 5^3 = 625

Out of them, we can only earn in six of them.

Let X be the player's gain in a single game.

E[X] = = 5000*1/216 + 1000*1/216 + 100*1/216 + 6*1/216 + 4*5/216 + 2*25/216 + 0

=> (5000+1000+100+6+20+50) / 216 = 6176/216 = 28.59

If a whole number answer is required, the closest value is 29.