Question

the table summarizes the movie - watching preferences and ages of a group of 500 people in one community
part a: determine p(dvd | age 18 - 40) and describe the event in everyday language. show all work. (2 points)
part b: are the events prefers dvd and being age 18 - 40 approximately independent? use probabilities to justify the answer. (2 points)

Explanation:

Part A:

Step1: Recall the formula for conditional probability

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). Here, \(A\) is the event of preferring DVD and \(B\) is the event of being in the age - 18 - 40 group.

Step2: Identify \(P(A\cap B)\) and \(P(B)\) from the table

From the table, \(P(\text{DVD}\cap\text{Age }18 - 40)=0.17\) and \(P(\text{Age }18 - 40)=0.17 + 0.17+0.06=0.4\)

Step3: Calculate \(P(\text{DVD}|\text{Age }18 - 40)\)

Substitute the values into the formula: \(P(\text{DVD}|\text{Age }18 - 40)=\frac{0.17}{0.4}=0.425\)
In everyday language, the probability that a person prefers DVDs given that they are in the 18 - 40 age group.

Part B:

Step1: Recall the formula for independence

Two events \(A\) (preferring DVD) and \(B\) (being in the 18 - 40 age group) are independent if \(P(A\cap B)=P(A)\times P(B)\)

Step2: Calculate \(P(A)\) and \(P(B)\)

\(P(\text{DVD})=0.17 + 0.06+0.01=0.24\), \(P(\text{Age }18 - 40)=0.4\)
\(P(A)\times P(B)=0.24\times0.4 = 0.096\)
From the table \(P(\text{DVD}\cap\text{Age }18 - 40)=0.17\)
Since \(0.17
eq0.096\), the events are not independent

Answer:

Part A:

\(P(\text{DVD}|\text{Age }18 - 40) = 0.425\). The probability that a person prefers DVDs given that they are in the 18 - 40 age group is \(0.425\)

Part B:

The events are not independent. Because \(P(\text{DVD}\cap\text{Age }18 - 40)=0.17\) and \(P(\text{DVD})\times P(\text{Age }18 - 40)=0.24\times0.4 = 0.096\), and \(0.17
eq0.096\)