Conditional probability question with set theory things

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SUMMARY

The discussion focuses on calculating conditional probabilities involving four independent events: A, B, C, and D, with probabilities P(A) = 0.04, P(B) = 0.03, P(C) = 0.02, and P(D) = 0.01. The user successfully computed the probability of at least one event occurring, denoted as P(F), which equals 0.09999976. The challenge lies in determining the probability of more than one event occurring given that at least one has occurred, represented as P(E | F), where E is defined as the union of pairs of events. The user seeks clarification on how to accurately express and calculate this probability.

PREREQUISITES
  • Understanding of basic probability concepts, including independent events
  • Familiarity with set theory notation and operations
  • Knowledge of conditional probability and its applications
  • Ability to manipulate and calculate probabilities using equations
NEXT STEPS
  • Study the concept of conditional probability in depth, focusing on the formula P(E | F) = P(E ∩ F) / P(F)
  • Learn about the inclusion-exclusion principle for calculating probabilities of unions of events
  • Explore examples of independent events and their implications in probability theory
  • Practice solving problems involving multiple independent events and conditional probabilities
USEFUL FOR

Students studying probability theory, mathematicians interested in set theory applications, and anyone looking to enhance their understanding of conditional probabilities in independent events.

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Homework Statement



Basically, I'm given the probability of 4 independent events:

P(A) = 0.04
P(B) = 0.03
P(C) = 0.02
P(D) = 0.01

If anyone of these occur, a failure will happen.
More than one can happen at the same time.

I need to find the probability that more than one of them has occurred given at least one as occurred.

Homework Equations





The Attempt at a Solution



I solved for the prob that at least one occurred:
Let F = A∪B∪C∪D
Then P(F) = P(A) + P(B) + P(C) + P(D) - P(A∩B∩C∩D) = 0.09999976

Now I have to solve for prob of 'more than one' occurred given F has occurred.
I'm stuck on this part.
How do I represent 'more than one occurred' with symbols?
Would it be E = (A∩B)∪(A∩C)∪(B∩D)∪(C∩D)?

Then P(E | F) ?
 
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No one..?
Is my post not understandable?
 

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