Probability Question - Picking a Ball

Question

There exist Two bags, one has one white ball, other has one black ball. You randomly select a bag and add a white ball in it. You then shake the bag. From the same bag, if you picked a ball and it was white, what is the chance that the other ball in that bag is also white?

I had the following idea: we have 2 cases

1. Balck Ball + White Ball or
2. White ball + White ball

probability of choosing either of the bag = 0.5

probability of picking a white ball = probability of picking from first case or from second = (0.5)(0.5) + (0.5)(1) = 0.75

probability of second ball being white = probability of second white ball in first case or second case = (0.5)(0) + (0.5)(1) = 0.5

so probability that other ball is also white = is 1/2.

Is this approach right?

Answer 1

$C1:$ You picked The Bag with White ball

Now $P(W)=1$ as there are all white balls in the bag.

$C2:$ You picked The Bag with Black ball

Now $P(W)=0.5$ as there is $1$ Black and $1$ White Ball in the bag.

Hence, $P(C1|W)=\frac{P(C1)P(W|C1)}{P(C1)P(W|C1)+P(C2)P(W|C2)}\Rightarrow P(C1|W)=\frac{(0.5)(1)}{(0.5)(1)+(0.5)(0.5)}=\frac{2}{3}$