Question

A group of people were asked what they drink during the day. They were asked whether they drink coffee, drink water, drink both coffee and water, or drink neither coffee nor water. The table shows the probabilities of the results. Let A represent drinking coffee and B represent drinking water. Which statement is true?

a) Drinking coffee and drinking water are not independent events because P(A|B)=P(A) and P(B|A)=P(B).
b) Drinking coffee and drinking water are independent events because P(A|B)=P(A) and P(B|A)=P(B).
c) Drinking coffee and drinking water are independent events because P(A|B)≠P(A) and P(B|A)≠P(B).
d) Drinking coffee and drinking water are not independent events because P(A|B)≠P(A) and P(B|A)≠P(B).

Answer 1

Drinking coffee and drinking water are not independent events because P(A|B)≠P(A) and P(B|A)≠P(B).

Step-by-step explanation:

Two events A and B are independent if the following conditions are true:

1. P(AB) = P(A)
2. P(A|B) = P(A)
3. P(B|A) = P(B)

Based on the information provided in the table, we can calculate the probabilities:

• P(A) = P(drinking coffee) = 0.3
• P(B) = P(drinking water) = 0.5
• P(AB) = P(drinking both coffee and water) = 0.15

Let's check if the conditions for independence hold:

1. P(AB) = P(A)P(B) = (0.3)(0.5) = 0.15, which is true.
2. P(A|B) = P(A) = 0.3, which is NOT equal to P(A). So, the first condition is NOT true.
3. P(B|A) = P(B) = 0.5, which is NOT equal to P(B). So, the second condition is NOT true.

Therefore, the statement that is true is d) Drinking coffee and drinking water are not independent events because P(A|B)≠P(A) and P(B|A)≠P(B).