what is the probability for a green marble to be drawn?

Question

Bag A contains 3 green marbles and 2 red marbles. Bag B contains 4 green marbles and 5 red marbles. choose one bag randomly and pick one marble from it. what is the probability for a green marble to be drawn?

Answer 1

Assuming that you are equally likely to pick Bag A as Bag B, here is how I would approach this problem. There are two cases we must consider:

1. You randomly select Bag A and draw a green from it
2. You randomly select Bag B and draw a green from it

Since these events are mutually exclusive, we can just add up the probabilities of these cases.

$$ P(\text{Drawing a green from Bag A})=\frac{1}{2} \times \frac{3}{5} = \frac{3}{10} \\ P(\text{Drawing a green from Bag B})=\frac{1}{2} \times \frac{4}{9} = \frac{4}{18} $$

Adding these two up gives us the answer:

$$ \frac{3}{10} + \frac{4}{18} = \frac{27}{90} + \frac{20}{90} = \frac{47}{90} $$

Answer 2

If I understood correctly:

$P(\text{Green}) = \left(\frac{1}{2} \times \frac{3}{5}\right)+\left(\frac{1}{2} \times \frac{4}{9}\right) = \frac{47}{90}$