Question

In a carnival game, a player spins a wheel that stops with the pointer on one (and only one) of three colors. The likelihood of the pointer landing on each color is as follows: 64 percent BLUE, 20 percent RED, and 16 percent GREEN.

Note: Your answers should be rounded to three decimal places.

(a) Suppose we spin the wheel, observe the color that the pointer stops on, and repeat the process until the pointer stops on BLUE. What is the probability that we will spin the wheel exactly three times?

(b) Suppose we spin the wheel, observe the color that the pointer stops on, and repeat the process until the pointer stops on RED. What is the probability that we will spin the wheel at least three times?

(c) Suppose we spin the wheel, observe the color that the pointer stops on, and repeat the process until the pointer stops on GREEN. What is the probability that we will spin the wheel 2 or fewer times?

Answer 1

Given:

In a carnival game, a player spins a wheel that stops with the pointer on one (and only one) of three colors. The likelihood of the pointer landing on each color is as follows:

BLUE = 64% = 0.64

RED = 20% = 0.20

GREEN = 16% = 0.16

Let X is a random variable shows the number of time we spin the wheel.

X follows the Geometric distribution.

The probability density function of geometric distribution is given by

P(X=x) = (1-p)^x-1 * p

a) Suppose we spin the wheel, observe the color that the pointer stops on, and repeat the process until the pointer stops on BLUE.

The probability that we will spin the wheel exactly three times is 0.083

P(X=3) = (1-0.64)^3-1 * 0.64

= (0.36)^2 * 0.64

= 0.083

P(X=3) = 0.083

b) Suppose we spin the wheel, observe the color that the pointer stops on, and repeat the process until the pointer stops on RED.

The probability that we will spin the wheel at least three times is 0.640

=P(red will not appear in first two times)

=(1-0.20)(1-0.20) = 0.640

c) Suppose we spin the wheel, observe the color that the pointer stops on, and repeat the process until the pointer stops on GREEN.

The probability that we will spin the wheel 2 or fewer times is 0.294

P(green appears on first spin+green does not appear on first*appear on second spin)

=0.16 +(1-0.16)*0.16 =

= 0.16 + 0.84 * 0.16

= 0.294