Question

This question has multiple parts.

A selection of soft-drink users is asked to taste the two disguised soft drinks and indicate which they prefer. The accompanying data contains the results of a simulated challenge on a college campus. Data listed below.

a. Determine the probability that a randomly chosen student prefers Brand A.
P(Brand A) =  (Do not use percents. Round to three decimal places as needed.)

b. Determine the probability that one of the students prefers Brand A and is less than 22 years old.
P(Brand A and Age < 22) =  (Do not use percents. Round to three decimal places as needed.)

c. Of those students who are less than 22 years old, calculate the probability that a randomly chosen student prefers (1) Brand A and (2) Brand B.

P(Brand A if Age < 22) =  (Do not use percents. Round to three decimal places as needed.)
P(Brand B if Age < 22) =  (Do not use percents. Round to three decimal places as needed.)

d. Of those students who are at least 22 years old, calculate the probability that a randomly chosen student prefers (1) Brand A and (2) Brand B.

P(Brand A if Age ≥ 22) =  (Do not use percents. Round to three decimal places as needed.)
P(Brand B if Age ≥ 22) =  (Do not use percents. Round to three decimal places as needed.)

Data:

Age	Preference
17	Brand A
18	Brand A
21	Brand A
18	Brand A
17	Brand A
23	Brand B
21	Brand B
20	Brand A
24	Brand B
21	Brand B
18	Brand A
18	Brand A
21	Brand B
21	Brand B
20	Brand B
20	Brand A
24	Brand A
18	Brand B
21	Brand B
21	Brand A
18	Brand B
18	Brand B
20	Brand B
21	Brand A
20	Brand B
18	Brand B
19	Brand A
21	Brand B
19	Brand A
23	Brand A
20	Brand B
24	Brand A
21	Brand B
18	Brand B
20	Brand B
23	Brand A
20	Brand B
23	Brand A
20	Brand B
17	Brand B
19	Brand B
18	Brand A
18	Brand A
21	Brand A
17	Brand B
19	Brand A
22	Brand B
21	Brand B
24	Brand A
17	Brand B
20	Brand B
21	Brand A
18	Brand A
20	Brand B
24	Brand B
16	Brand A
23	Brand A
20	Brand A
21	Brand B
24	Brand A
19	Brand A
23	Brand B
18	Brand A
23	Brand B
20	Brand A
23	Brand A
21	Brand B
21	Brand A
19	Brand B
22	Brand B
19	Brand A
20	Brand A
21	Brand A
22	Brand A
23	Brand B
21	Brand B
21	Brand A
18	Brand B
18	Brand A
21	Brand B
16	Brand B
21	Brand B
21	Brand B
18	Brand A
22	Brand B
22	Brand B
17	Brand A
23	Brand B
17	Brand A
17	Brand A
22	Brand A
19	Brand B
21	Brand B
22	Brand A
17	Brand B
18	Brand B
23	Brand A
20	Brand A
23	Brand A

Answer 1

The given data is transformed into an Age vs Brand Preference Frequency distribution as:

Age	Brand A	Brand B	Total
16	1	1	2
17	5	4	9
18	10	7	17
19	5	3	8
20	6	9	15
21	8	15	23
22	3	4	7
23	7	5	12
24	4	2	6
Total	49	50	99

a) The probability that a person prefers Brand A is computed here as:
P(Brand A)
= n(Brand A) / n(Total)

= 49/99 = 0.495
Therefore 0.495 is the required probability here.

b) P(Brand A and Age < 22 years)

= n(Brand A and Age < 22 years) / n(Total)

= (49 - 3 - 7 - 4) / 99

= 35/99

= 0.354

Therefore 0.354 is the required probability here.

c) n( < 22 Age) = 99 - (6 + 12 + 7) = 74

Using Bayes theorem, we get here:
P(Brand A if Age < 22) = P(Brand A and Age < 22) / P(Age < 22)

= 35 / 74

= 0.473

Therefore 0.473 is the required probability here.

P(Brand B if Age < 22) = 1 - P(Brand A if Age < 22) = 1 - 0.473 = 0.527

Therefore 0.527 is the required probability here.

d) n(Age >= 22) = n(Total) - n(Age < 22) = 99 - 74 = 25

Also n(Brand A and Age >= 22) = 3 + 7 + 4 = 14

Therefore,
P(Brand A if Age >= 22) = 14/25 = 0.56
Therefore 0.56 is the required probability here.

P(Brand B if Age >= 22) = 1 - 0.56 = 0.44
Therefore 0.44 is the required probability here.