union sets probability

Contents

	• one choice
	• many choices: permutations
	• many choices: combinations
	• many choices: combinations
	• independent events
	• dependent events
	• permutations without repettitions
	• permutations with identical elements
	• combinations
	• coditional probabilities
	• and or events
	• two outcomes
	• total probability theorem
	• Bayes rule
	• union sets and probability
	• fundamental counting principle

Other exercices

Combinatorics - Probability

union sets probability
probability with union sets

We have:
A ∪ B = (X ∩ A) ∪ (A ∩ B) ∪ (Y ∩ B)   (1)
Then:
P(A ∪ B) = P(X ∩ A) + P(A ∩ B) + P(Y ∩ B)   (1')

A = (X ∩ A) ∪ (A ∩ B)   (2)
B = (Y ∩ B) ∪ (A ∩ B)   (3)

P(A) = P(X ∩ A) + P(A ∩ B)   (2')
P(B) = P(Y ∩ B) + P(A ∩ B)   (3')

P(X ∩ A) = P(A) - P(A ∩ B)    (2")
P(Y ∩ B) = P(B) - P(A ∩ B)   (3")

P(A ∪ B) = P(X ∩ A) + P(A ∩ B) + P(Y ∩ B) (1")
= P(A) - P(A ∩ B) + P(A ∩ B) + P(B) - P(A ∩ B)
= P(A) + P(B) - P(A ∩ B)

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Today: :
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