# Bag of Marbles and Law of Total Probability

There are 3 bags each containing 100 marbles. Bag 1 has 75 red and 25 blue marbles. Bag 2 has 60 red and 40 blue marbles. Bag 3 has 45 red and 55 blue marbles. Now a bag is chosen at random and a marble is also picked at random.

1) What is the probability that the marble is blue?

2) What happens when the ﬁrst bag is chosen with probability 0.5 and other bags with equal probability each?

I understand the #1, I got .4 but how would I approach doing #2?

Thank you

You do it just like you did the first part, only instead of saying that the probability of drawing each bag is $$\frac13$$, you use the given probabilities. For example, the probability of draw a blue marble from bag $$1$$ is $$\frac12\cdot\frac14=\frac18$$ because there is a probability of $$\frac12$$ that you pick bag $$1$$, and then a probability of $$\frac14$$ that you draw a blue marble from it. Do the same sort of thing for the other two bags, and add up the probabilities.