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: 65 percent BLUE, 22 percent RED, and 13 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?
a: pr=(notgettingblue)(notgetting blue)(gettingblue)=.78^2*.22
dik
0.866
To answer these questions, we can use the concept of geometric probability. Geometric probability deals with the probability of a certain event occurring for the first time after a fixed number of independent trials.
(a) To find the probability that we will spin the wheel exactly three times until it stops on BLUE, we need to find the probability of the first two spins landing on non-blue colors, followed by a blue on the third spin.
The probability of not getting blue on the first two spins is calculated by multiplying the probabilities of spinning non-blue colors together:
(1 - 0.65) * (1 - 0.65) = 0.35 * 0.35 = 0.1225
The probability of getting blue on the third spin is simply the probability of getting blue, which is 0.65.
To find the probability of spinning the wheel exactly three times until it stops on BLUE, we multiply the probabilities calculated above:
0.1225 * 0.65 = 0.0796
Therefore, the probability of spinning the wheel exactly three times until it stops on BLUE is approximately 0.080.
(b) To find the probability that we will spin the wheel at least three times until it stops on RED, we need to find the probability of not getting red on the first two spins, followed by getting red on the third spin or any subsequent spins.
The probability of not getting red on the first two spins is calculated by multiplying the probabilities of spinning non-red colors together:
(1 - 0.22) * (1 - 0.22) = 0.78 * 0.78 = 0.6084
Therefore, the probability of getting red on the third spin or any subsequent spins is 1 - the probability of not getting red on the first two spins:
1 - 0.6084 = 0.3916
Therefore, the probability of spinning the wheel at least three times until it stops on RED is approximately 0.392.
(c) To find the probability that we will spin the wheel 2 or fewer times until it stops on GREEN, we need to find the probability of getting green on the first spin or the second spin.
The probability of getting green on the first spin is simply the probability of getting green, which is 0.13.
The probability of not getting green on the first spin and then getting green on the second spin is calculated by multiplying the probabilities of spinning non-green colors on each of the two spins:
(1 - 0.13) * (1 - 0.13) = 0.87 * 0.87 = 0.7569
Therefore, to find the probability of spinning the wheel 2 or fewer times until it stops on GREEN, we add the probabilities calculated above:
0.13 + 0.7569 = 0.8869
Therefore, the probability of spinning the wheel 2 or fewer times until it stops on GREEN is approximately 0.887.