Answer to Question #154617 in Statistics and Probability for asia

Question

An experiment is run in the following manner. The colors red, yellow, and blue are each flashed on a screen for a short period of time. A subject views the colors and is asked to choose the one he feels was flashed for the longest amount of time. The experiment is repeated three times with the same subject. If all the colors were flashed for the same length of time, give the probability distribution for y, the number of times the subject choose the color red. Assume that his three choice are independent.

Answer 1

There are three colors and all are equally likely so each of three times subject can chose one out of three colors. So, possible number of outcomes in the sample space is

$3 \times 3 \times 3 = 27$

Since x shows the number of times subject chose red out of three. Here x can take values 0, 1, 2, and 3. When x=0, then subjects chose either yellow or blue color. There number of outcomes in favor of this event is

$2 \times 2 \times 2 = 8$

The required probability is

$P(X=0) = \frac{8}{27}$

When x=1, then subject chose either yellow or blue color 2 times, red color one time. Number of ways of selecting red color is $C^3_1 = 3$ . The possible number of outcomes in favor of this event is

$3 \times 2 \times 2 = 12$

The required probability is

$P(X=1) = \frac{12}{27}$

When x=2, then subject chose either yellow or blue color 1 times, red color 2 times. Number of ways of selecting red color is $C^3_2=3$ . then subject chose either yellow or blue color 1 times, red color 2 times. Number of ways of selecting red color is

$3 \times 2 = 6$

The required probability is

$P(X=2) = \frac{6}{27}$

When x=3, then subject chose red color 3 times. The possible number of outcomes in favor of this event is

$1 \times 1 \times 1 = 1$

The required probability is

$P(X=3) = \frac{1}{27}$

The probability distribution of X: