Question

The conditional relative frequency table was calculated by row using data from a survey of one station’s television programming. The survey compared the target audience with the type of show, either live or recorded, over a 24-hour time period.

Which is the best description of the value 0.375 in the table?

Given that the program was targeted at adults, there is a 37.5% chance that it was recorded.
Given that the program was recorded, there is a 37.5% chance that it was targeted at adults.
37.5% of the programs are targeted at adults.
37.5% of the programs are recorded.

Answer 1

Given that the program was targeted at adults, there is a 37.5% chance that it was recorded.

Explanation:

The best description is that:

It was given that the program was targeted so there is a 37.5% chance that it was recorded.

Since the value 0.375 is obtained when the frequency of the targeted adults with the recorded show were divided by the total frequency of the adults.

Hence, we represent this obtained frequency in percentage as:

0.375×100=37.5%

i.e.

Let A denote the event that the targeted audience was adult

and B denote the event that the show is recorded.

Let P denote the probability of an event.

So, we have:

`P(B|A)=(P(A\bigcap B))/(P(A))=0.375=37.5\%`

Answer 2

The correct option would be-

Given that the program was targeted at adults, there is a 37.5% chance that it was recorded.

Solution-

We know that, probability of A given that B is,

`P(A|B)=(P(A\ \cup\ B))/(P(B))`

From the table,

`P(\text{Recorded})=0.417\\\\P(\text{Adult}) = 1\\\\P(\text{Recorded}\ |\ \text{Adult})=0.375`

Probability that the program was recorded given that the targeted audiences were adults is,

`P(\text{Recorded}\ |\ \text{Adult})=\frac{P(\text{Recorded}\ \cup\ \text{Adult})}{P(\text{Adult})}`

`(0.375)/(1)=0.375=37.5\%`

Probability that the targeted audiences were adults given that the program was recorded is,

`P(\text{Adult}\ |\ \text{Recorded})=\frac{P(\text{Adult}\ \cup\ \text{Recorded})}{P(\text{Recorded})}`

`(0.375)/(0.417)=0.899=89.9\%`

Therefore, the first option "Given that the program was targeted at adults, there is a 37.5% chance that it was recorded." is correct