The maximum percentage of people who have studied neither $A$ nor $B$

Question

In school $X$, $40\%$ of students have studied a language at some point during their school education, and 20% have studied history at some point in their school education. What is the maximum percentage of people who have studied neither?

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I was thinking:

All = $100\%$

$P(L) =$ language ; $P(H)$ = history

$P(L) + P(H) – P(L \cap H) + P(\text{neither}) = 100\% $

$0.4 + 0.2 – P(L \cap H) + P(\text{neither}) = 1$

$P(\text{neither}) = 0.4 + P(L \cap H)$

So $P(\text{neither})$ is max when $P(L \cap H)$ is max. That happens when all the ones studying $H$ also study $L$. So that would be 20%

$P(\text{neither}) = 0.4 + 0.2 = 0.6$

What do you think?

Answer 1

Your working before this step is all good:

$P(\text{neither}) = 0.4 + P(L \cap H)$

The minimum value of $P(L \cap H)$ leads to the maximum value of $P(\text{neither})$.

Correction:

The maximum value of $P(L \cap H)$ leads to the maximum value of $P(\text{neither}).$

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Alternatively, visualise a Venn diagram:

We want to maximise the number of outcomes in the shaded region. As such, encase event $H$ entirely within event $L:$