Question

A survey of 1,200 men and women asked, "Do you earn over $75,000 per year?" The table below shows the responses for males and females:
Male Female Total
Income over $75,000 585 485 1,070
Income below $75,000 65 65 130
Total 650 550 1200


Based on these data, are "being female" and "earning over $75,000" independent events?
No, P(being female | the person earns over $75,000) = P(being female)
No, P(being female | the person earns over $75,000) ≠ P(being female)
Yes, P(being female | the person earns over $75,000) = P(being female)
Yes, P(being female | the person earns over $75,000) ≠ P(being female)

Answer 1

Two events are said to be independent of each other, when the probability that one event occurs in no way affects the probability of the other event occurring. For example, the outcome of rolling a die has no way of affecting the outcome of flipping a coin. Thus we say that both events are indipendent events.

In conditional probability, in the case where events A and B are independent, the conditional probability of event B given event A, P(B|A) is simply the probability of event B, that is P(B).

Recall that the conditional probability of event, say B given event, say A, P(B|A) is given by
$P(B|A)= \frac{P(B\cap A)}{P(A)}$

Given the table
                                           Male      Female       Total
Income over $75,000          585          485          1,070
Income below $75,000        65            65            130
Total                                    650          550          1,200

P(being female | the person earns over $75,000) = P(Females that earns over $75,000) divided by P(people that earns over $75,000).

P(Females that earns over $75,000) = $\frac{485}{1200} = \frac{97}{240}$

P(people that earns over $75,000) = $\frac{1070}{1200} = \frac{107}{120}$

Thus, P(being female | the person earns over $75,000) = $\frac{\frac{97}{240}}{\frac{107}{120}} = \frac{97}{214}$

P(being female) = $\frac{550}{1200} = \frac{11}{24}$

Therefore, since P(being female | the person earns over $75,000) is not equal to P(being female), then "being female" and "earning over $75,000" are NOT independent events.

Therefore, the correct answer is
No, P(being female | the person earns over $75,000) ≠ P(being female)