There are two types of apples – red and green. The ratio of red to green is $4:1$. They are mixed together before they are boxed. There is $\frac{1}{50}$ chance that a green apple is discarded, and $\frac{1}{100}$ chance that a red apple is discarded. What is the probability that an apple selected will be discarded?
Attempt: I have
Green apple discarded OR red apple discarded
$\frac{4}{5}\frac{1}{50}$ + $\frac{1}{5}\frac{1}{100}$
Question: Why is this incorrect? It seems I have to multiply by $\frac{2}{3}$ to get the correct answer.
Best Answer
You seem to have confused red and green and your calculation should have been $$\frac{4}{5}\frac{1}{100}+\frac{1}{5}\frac{1}{50}=\frac{3}{250}$$
As a check, suppose you had $1000$ apples
You would expect to have $800$ red apples and $200$ green apples
You would expect to discard $8$ red apples and $4$ green apples
So you would expect to discard $12$ apples, $\frac{12}{1000}=\frac{3}{250}$ of the total