Question 1197312: Two hundred consumers are surveyed about a new brand of snack food, Crunchicles. Their age groups and preferences are given in the table.
18 - 24 25 - 34 35 - 55 55 and over Total
Liked Crunchicles 3 11 7 27 48
Disliked Crunchicles 10 9 3 12 34
No Preference 17 17 11 73 118
Total 30 37 21 112 200
One consumer from the survey is selected at random. Leave all answers in a reduced fraction.
What is the probability that the consumer is 18 - 24 years of age, given that he/she dislikes Crunchicles?
What is the probability that the selected consumer dislikes Crunchicles?
What is the probability that the selected consumer is 35 - 55 years old or likes Crunchicles?
If the selected consumer is 70 years old, what is the probability that he/she likes Crunchicles?
Found 2 solutions by ewatrrr, math_tutor2020:
Answer by ewatrrr(24785)   (Show Source):
Answer by math_tutor2020(3817)  (Show Source):
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Given Data Table:
| Age Group | | | 18 - 24 | 25 - 34 | 35 - 55 | 55 and over | Total | | Liked Crunchicles | 3 | 11 | 7 | 27 | 48 | | Disliked Crunchicles | 10 | 9 | 3 | 12 | 34 | | No Preference | 17 | 17 | 11 | 73 | 118 | | Total | 30 | 37 | 21 | 112 | 200 |
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Question: What is the probability that the consumer is 18 - 24 years of age, given that he/she dislikes Crunchicles?
"Given that he/she dislikes Crunchicles" means we know 100% the person doesn't like snack.
That means we don't have to worry about the "liked" and "no preference" rows. We focus entirely on "disliked".
In this row, there are 10 such people between ages 18-24, out of 34 people total.
Divide the values to form a fraction. Reduce as much as possible.
10/34 = (2*5)/(2*17) = 5/17
Answer: 5/17
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Question: What is the probability that the selected consumer dislikes Crunchicles?
This time we consider everyone.
There are 34 people who disliked Crunchicles out of 200 total people.
34/200 = (2*17)/(2*100) = 17/100
Answer: 17/100
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Question: What is the probability that the selected consumer is 35 - 55 years old or likes Crunchicles?
Define these three events
A = person is in the age range 35-55
B = person likes Crunchicles
C = number of people who are BOTH in the age range 35-55 AND also like Crunchicles
In terms of a Venn Diagram, region C is the overlapped portion of circles A and B.
According to the table
n(A) = number of people in age range 35-55
n(A) = 21
n(B) = 48
n(C) = 7
Then we use the inclusion-exclusion principle
n(A or B) = n(A) + n(B) - n(A and B)
n(A or B) = 21 + 48 - 7
n(A or B) = 62
There are 62 people who are in the age range 35-55, or they liked Crunchicles, or both.
The subtraction of n(A and B) is to correct for double-counting of people in region C.
This is out of the 200 total people surveyed.
62/200 = (2*31)/(2*100) = 31/100
Answer: 31/100
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Question: If the selected consumer is 70 years old, what is the probability that he/she likes Crunchicles?
We focus on the "55 and over" column because we know the person is 70 years old, which of course fits the "55 and over" age range.
Ignore any other column.
There are 27 such people who liked the snack out of 112 total in this column.
It forms the probability 27/112 which cannot be reduced further.
Note that each prime factorization is
27 = 3*3*3
112 = 2*2*2*2*7
there aren't any common factors other than 1.
Answer: 27/112
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