I have two bowls, Bowl A and Bowl B. Bowl A has 1 blue marble, and 4 yellow marbles. Bowl B has 3 blue and 2 yellow marbles.
You randomly chose a bowl, and randomly pick a marble. This marble is blue. What is the probability you picked from bowl A, given that you have picked a blue marble?
Initially I used the conditional probability formula as follows:
$$P(Bowl \, A| Picking\,Blue\, Marble) = \frac{P(Bowl\,A \cap Picking\,Blue\, Marble)} {P(Picking\,Blue\, Marble)} = \frac{\frac{1}{5}}{\frac{4}{10}} = \frac{1}{2}$$
My reasoning was that picking Bowl A and Picking a Blue marble is 1/2*4/10 = 1/5. And the probability of picking a blue marble from either bowl is 1/4.
However, the answer is a bit more intuitive. Without the formula it is simply 1/4. What have I done wrong?
Best Answer
Hint: Your probability for the intersection is not right. The intersection can be expressed with the conditional probability.
$P(\text{Bowl A} \cap \text{Picking blue marble})=P(\text{Bowl A})\cdot P(\text{Picking blue marble|Bowl A})$
$P(\text{Picking blue marble|Bowl A})$: Probability of picking a blue marble, given it is drawn from Bowl A.
I think you can figure out the values.