How do you find p(b) when p(a) is 23 p(ba) is 12 and p(a U b) is 45 and is a dependent event?

Answer 1

There are symbols missing from your question which I cam struggling to guess and re-insert.

p(a) = 2/3

p(b ??? a) = 1/2

p(a ∪ b) = 4/5

p(b) = ?

Why use the set notation of Union on the third given probability whereas the second probability has something missing but the "sets" are in the other order, and the order wouldn't matter in sets.

There are two possibilities:

1) The second probability is: p(b ∩ a) = p(a ∩ b) = 1/2

→ p(a) + p(b) = p(a ∪ b) + p(a ∩ b)

→ p(b) = p(a ∪ b) + p(a ∩ b) - p(a)

= 4/5 + 1/2 - 2/3

= 24/30 + 15/30 - 20/30

= 19/30

2) The second and third probabilities are probabilities of "given that", ie:

p(b|a) = 1/2

p(a|b) = 4/5

→ Use Bayes theorem:

p(b)p(a|b) = p(a)p(b|a)

→ p(b) = (p(a)p(b|a))/p(a|b)

= (2/3 × 1/2) / (4/5)

= 2/3 × 1/2 × 5/4 = 5/12