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Conditional Probabilities

"The events A and B are such that P(A|B) = 7/10, P(B|A) = 7/15 and P(A U B) = 3/5. Find the values of P(A ∩ B) and P(A’ ∩ B)."

Anyone know tf how to do this, my mind gone blank?? Pretty sure i need the P(A) first to put into a formula for P(AnB) = P(A) X 7/15, so how do i find P(A)?
Original post by isobel_07
"The events A and B are such that P(A|B) = 7/10, P(B|A) = 7/15 and P(A U B) = 3/5. Find the values of P(A ∩ B) and P(A’ ∩ B)."

Anyone know tf how to do this, my mind gone blank?? Pretty sure i need the P(A) first to put into a formula for P(AnB) = P(A) X 7/15, so how do i find P(A)?


P(A∣B)=P(A∩B)P(B)=710P(A|B) = \dfrac{P(A \cap B)}{P(B)} = \dfrac{7}{10} .... (*)

P(B∣A)=P(A∩B)P(A)=715P(B|A) = \dfrac{P(A \cap B)}{P(A)} = \dfrac{7}{15} .... (**)

P(A∪B)=P(A)+P(B)−P(A∩B)=35P(A \cup B) = P(A) + P(B) - P(A \cap B) = \dfrac{3}{5} .... (***)

You can express P(B)P(B) in terms of P(A∩B)P(A \cap B) from (*) and also express P(A)P(A) in terms of P(A∩B)P (A \cap B) from (**) then sub these into (***) and solve for P(A∩B)P(A\cap B).

Then notice that P(B)=P(A′∩B)+P(A∩B)P(B) = P(A ' \cap B) + P(A \cap B) hence you can use part (*) as well as the value of P(A∩B)P(A \cap B) to work out P(A′∩B)P(A ' \cap B)

At no point do you need to find P(A)P(A), believe it or not.
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isobel_07
OP
12
Original post by RDKGames
P(A∣B)=P(A∩B)P(B)=710P(A|B) = \dfrac{P(A \cap B)}{P(B)} = \dfrac{7}{10} .... (*)

P(B∣A)=P(A∩B)P(A)=715P(B|A) = \dfrac{P(A \cap B)}{P(A)} = \dfrac{7}{15} .... (**)

P(A∪B)=P(A)+P(B)−P(A∩B)=35P(A \cup B) = P(A) + P(B) - P(A \cap B) = \dfrac{3}{5} .... (***)

You can express P(B)P(B) in terms of P(A∩B)P(A \cap B) from (*) and also express P(A)P(A) in terms of P(A∩B)P (A \cap B) from (**) then sub these into (***) and solve for P(A∩B)P(A\cap B).

Then notice that P(B)=P(A′∩B)+P(A∩B)P(B) = P(A ' \cap B) + P(A \cap B) hence you can use part (*) as well as the value of P(A∩B)P(A \cap B) to work out P(A′∩B)P(A ' \cap B)

At no point do you need to find P(A)P(A), believe it or not.


ah, i see was overthinking it a bit, thank.

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