Question

A three-question quiz has the following possible choice options

Question 1: (A, B, C)
Question 2: (T, F}
Question 3: (T, F)
The sample space is (ATT, ATF, AFT, AFF, BTT, BTF, BFT, BFF, CTT, CTF, CFT, CFF).
Let X be the number of times a "T" is chosen. Construct a probability distribution for X.
X
0
1
2
3
P(X)

Answer 1

To construct the probability distribution for X, we need to determine the probability of each possible value of X occurring.

Let's go through each possible value of X and calculate its probability:

X = 0: This means that "T" is not chosen at all. There are 4 outcomes in the sample space that satisfy this condition: AFF, BFF, CFF. Therefore, the probability of X = 0 is 4/12 = 1/3.

X = 1: This means that "T" is chosen once. There are 6 outcomes in the sample space that satisfy this condition: ATF, AFT, BTF, BFT, CTF, CFT. Therefore, the probability of X = 1 is 6/12 = 1/2.

X = 2: This means that "T" is chosen twice. There is only 1 outcome in the sample space that satisfies this condition: ATT. Therefore, the probability of X = 2 is 1/12.

X = 3: This means that "T" is chosen three times. There are no outcomes in the sample space that satisfy this condition. Therefore, the probability of X = 3 is 0.

Now, we can summarize the probability distribution for X:

X | Probability

0 | 1/3 1 | 1/2 2 | 1/12 3 | 0

Please note that the probabilities should add up to 1, which is the case in this probability distribution.