# Confusing probabilities with inclusion-exclusion

Hello,

I have a question about this practice problem. The explanation of the solution to the problem given on that page makes intuitive sense to me, but I got to a different solution and can't see where I went wrong.

The problem concerns a selection of students with the probability of each student being enrolled in

• French being $P(F)=0.6$
• Spanish being $P(S)=0.3$
• neither being $P(\overline{F \cup S})=0.2$.

It asks for the probability of one student being in French but not in Spanish $P(F\backslash S) = P(Q)$.

So my understanding and this explainer leads me to believe, that $P(F \cap S) = P(B) = P(F) \cdot P(S) = 0.18. \$/extract_itex] Also, I was quite confident that $P(Q) = P(F \backslash B) = P(F)-P(B)=0.42. \\$ But that is of course different from the intuitively explained solution of $P(Q)=0.5$. What was my mistake? Note by Christian Grosser 5 months ago This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science. When posting on Brilliant: • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused . • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone. • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge. • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events. MarkdownAppears as *italics* or _italics_ italics **bold** or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote  # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in $$ ... $$ or \[ ... $ to ensure proper formatting.
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The probability of a student studying french or spanish is not mutually exclusive, so $P(B) = P(F) \cdot P(S)$ does not hold.

It should

$P(B) = (P \cup S) - P(S) = (1 - 0.8) - 0.3 = 0.5$.

- 5 months ago

$P(B) = P(F \cap S)$ is not $0.5$, $P(F \backslash S)$ is. You likely mean $P(F \backslash S) = P(F \cup S) - P(S) = 1-0.2-0.3=0.5$. But that clears things up. Thank you.

- 5 months ago