The table summarizes the movie-watching preferences and ages of a group of 500 people in one community.Movie Theater Streaming Service DVDUnder age 18 0.13 0.16 0.01Age 18–40 0.17 0.17 0.06Over Age 40 0.13 0.03 0.14Part A: Determine P(streaming | under age 18) and describe the event in everyday language. Show all work. (2 points)Part B: Are the events prefers streaming service and being under age 18 approximately independent? Use probabilities to justify the answer. (2 points)
Question
The table summarizes the movie-watching preferences and ages of a group of 500 people in one community.Movie Theater Streaming Service DVDUnder age 18 0.13 0.16 0.01Age 18–40 0.17 0.17 0.06Over Age 40 0.13 0.03 0.14Part A: Determine P(streaming | under age 18) and describe the event in everyday language. Show all work. (2 points)Part B: Are the events prefers streaming service and being under age 18 approximately independent? Use probabilities to justify the answer. (2 points)
Solution
Part A:
P(streaming | under age 18) refers to the probability of a person using a streaming service given that they are under the age of 18.
From the table, we can see that the probability of a person being under 18 and using a streaming service is 0.16.
So, P(streaming | under age 18) = 0.16
In everyday language, this means that if you randomly select a person from this community who is under the age of 18, there is a 16% chance that they prefer to watch movies using a streaming service.
Part B:
Two events are independent if the probability of both events occurring is the product of the probabilities of each event occurring individually.
Let's denote: P(A) = Probability of a person being under age 18 = 0.13 + 0.16 + 0.01 = 0.30 P(B) = Probability of a person preferring streaming service = 0.16 + 0.17 + 0.03 = 0.36 P(A ∩ B) = Probability of a person being under age 18 and preferring streaming service = 0.16
If A and B were independent, we would expect P(A ∩ B) = P(A) * P(B).
Let's check this:
P(A) * P(B) = 0.30 * 0.36 = 0.108
Since P(A ∩ B) = 0.16 ≠ P(A) * P(B), the events are not independent.
So, the events of preferring a streaming service and being under age 18 are not approximately independent in this community.
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