Discarding apples - why is my reasoning wrong? 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.

Haven't you mixed it up a bit?
Let $P(G)$ be probability of a green apple being selected and $P(R)$ be probability of a red apple being selected.
Let $P(D|G)$ be probability of discard if green apple and $P(D|R)$ be probability of discard if red apple.
Probability of an apple being discarded $P(D) = P(G)\cdot P(D|G) + P(R)\cdot P(D|R) = \frac 15 \frac 1{50} + \frac 45 \cdot \frac 1{100} = \frac 3{250}$
