Confusing probabilities with inclusion-exclusion


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

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

So my understanding and this explainer leads me to believe, that P(FS)=P(B)=P(F)P(S)=0.18.P(F \cap S) = P(B) = P(F) \cdot P(S) = 0.18. \\ Also, I was quite confident that P(Q)=P(F\B)=P(F)P(B)=0.42. 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)=0.5P(Q)=0.5.

What was my mistake?

Note by Christian Grosser
7 months, 3 weeks ago

No vote yet
1 vote

  Easy Math Editor

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. numbered
2. 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 1

paragraph 2

paragraph 1

paragraph 2

[example link]( 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.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


Sort by:

Top Newest

The probability of a student studying french or spanish is not mutually exclusive, so P(B)=P(F)P(S)P(B) = P(F) \cdot P(S) does not hold.

It should

P(B)=(PS)P(S)=(10.8)0.3=0.5P(B) = (P \cup S) - P(S) = (1 - 0.8) - 0.3 = 0.5 .

Pi Han Goh - 7 months, 3 weeks ago

Log in to reply

P(B)=P(FS)P(B) = P(F \cap S) is not 0.50.5, P(F\S)P(F \backslash S) is. You likely mean P(F\S)=P(FS)P(S)=10.20.3=0.5 P(F \backslash S) = P(F \cup S) - P(S) = 1-0.2-0.3=0.5 . But that clears things up. Thank you.

Christian Grosser - 7 months, 3 weeks ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...