Question

A bag contains 1 blue, 2 green, and 2
red marbles. You choose a marble with one hand and then a second marble with the other hand.
What is the probability that you will first choose a red marble followed by a green marble?

Answer 1

To find the probability of first choosing a red marble followed by a green marble from the bag, we can break the problem into two parts: the probability of choosing a red marble first, and then the probability of choosing a green marble second.

First, we calculate the total number of marbles in the bag:

- 1 blue
- 2 green
- 2 red

Total marbles = 1 + 2 + 2 = 5 marbles.

### Step 1: Probability of choosing a red marble first

The probability of selecting a red marble first is calculated as follows:

\[
P(\text{Red first}) = \frac{\text{Number of red marbles}}{\text{Total number of marbles}} = \frac{2}{5}
\]

### Step 2: Probability of choosing a green marble second

After removing one red marble (since we are choosing marbles without replacement), we now have:

- 1 blue
- 2 green
- 1 red

Total remaining marbles = 1 + 2 + 1 = 4 marbles.

Now we can calculate the probability of selecting a green marble second:

\[
P(\text{Green second | Red first}) = \frac{\text{Number of green marbles remaining}}{\text{Total remaining marbles}} = \frac{2}{4} = \frac{1}{2}
\]

### Step 3: Combined Probability

Now, we can find the overall probability of first choosing a red marble and then a green marble by multiplying the probabilities from the two steps:

\[
P(\text{Red first and Green second}) = P(\text{Red first}) \times P(\text{Green second | Red first}) = \frac{2}{5} \times \frac{1}{2} = \frac{2}{10} = \frac{1}{5}
\]

Thus, the probability that you will first choose a red marble followed by a green marble is

\[
\boxed{\frac{1}{5}}.
\]