Question

A large study of German children of Caucasian descent found that 46% have brown hair and that 25% have freckles. Nonetheless, you cannot conclude that because (0.46)(0.25)=0.115 about 11.5% of German children of Caucasian descent are brown‑haired with freckles. Why not? Having brown hair and having freckles are definitely independent events. Having brown hair and having freckles are not necessarily independent events. Having brown hair and having freckles are definitely disjoint events. Having brown hair and having freckles are not necessarily disjoint events.

Answer 1

Having brown hair and having freckles are not necessarily independent events

Explanation:

We are given the following information in the question:

Percentage of children having brown hairs = 46%

Percentage of people children having freckles = 25%

But it cannot be concluded that

German children of Caucasian descent are brown‑haired with freckles =

`0.46* 0.25 = 0.115 = 11.5\%`

This is not always true because of the following reasons:

Brown hair and having freckles are not necessarily independent events.

We can not assume that event of a child having brown hair is independent of the event of child having freckles. The events could be dependent as well.

The above will only be true when event of a child having brown hair is independent of the event of child having freckles.

If the events are independent then,

`P(A \cap B) = P(A)* P(B)`