Conditional probabilities exercises

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I've been practicing some conditional probabilities and got stuck on a few of them

  1. Find $P(B^{\prime}|A^{\prime})$, if $P(A) = a$, $P(B) = b$ and $P(A \cap B) = c$. ($a = 0.48$, $b = 0.33$, $c = 0.17$)
  2. Urn contains: $k$ white balls, $m$ red balls and $n$ black balls. without returning the balls are taken out of the urn. what is the probability of getting a white ball earlier than getting a black ball ($k = 5$, $m = 2$, $n = 7$)
  3. events $A$ and $B$ probabilities are $a$ and $b$ respectively, the probability for one of these events to happen is $c$. it is known that event $B$ has happened, what is the probability of event $A$ happening as well.

i have an idea how to solve #1, but have no clue on the rest of them. i am not as much interested in answers as i am in solutions, so any input will be appreciated

Best Answer

I found question 3 easier so answering it first.

You are given $P(A) = a$ and $P(B) = b$ and probability for occurrence of any of the given events is c i.e. $P(A \cup B) = c$ and you are to find $P(A|B) =\:?$

So, $P(A|B) = \frac{P(A \cap B)}{P(B)}$

Here you are not given that A and B are independent events so $P(A\cap B) \ne P(A).P(B)$

So taking $P(A \cap B) = P(A) + P(B) - P(A \cup B) = a + b - c$

$\therefore P(A|B) = \frac{a+b-c}{b}$

Above is my opinion and it seems correct at least to me. Do let me know if you don't feel good about it.

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