There are 7 orange disks and 5 green disks in bag X and there are 5 orange disks and 15 green disks in bag Y. If one disk is selected at random from each bag, what is the probability that both disks selected are green ?

Options

A. `(5)/(48)`
B. `(7)/(48)`
C. `(5)/(16)`
D. `(7)/(16)`

Explanation

To find the probability that both disks selected from Bag X and Bag Y are green, we can follow these steps:

### Step 1: Determine the total number of disks in each bag.
- **Bag X**: There are 7 orange disks and 5 green disks.
- Total disks in Bag X = 7 + 5 = 12 disks.

- **Bag Y**: There are 5 orange disks and 15 green disks.
- Total disks in Bag Y = 5 + 15 = 20 disks.

### Step 2: Calculate the probability of selecting a green disk from Bag X.
- The number of green disks in Bag X = 5.
- The total number of disks in Bag X = 12.

The probability of selecting a green disk from Bag X is:
\[ P(\text{Green from X}) = \frac{\text{Number of green disks in X}}{\text{Total disks in X}} = \frac{5}{12} \]

### Step 3: Calculate the probability of selecting a green disk from Bag Y.
- The number of green disks in Bag Y = 15.
- The total number of disks in Bag Y = 20.

The probability of selecting a green disk from Bag Y is:
\[ P(\text{Green from Y}) = \frac{\text{Number of green disks in Y}}{\text{Total disks in Y}} = \frac{15}{20} = \frac{3}{4} \]

### Step 4: Calculate the combined probability of both events.
Since the selection of disks from Bag X and Bag Y are independent events, we can multiply their probabilities:
\[ P(\text{Both green}) = P(\text{Green from X}) \times P(\text{Green from Y}) = \frac{5}{12} \times \frac{3}{4} \]

### Step 5: Perform the multiplication.
\[ P(\text{Both green}) = \frac{5 \times 3}{12 \times 4} = \frac{15}{48} \]

### Step 6: Simplify the fraction.
To simplify \(\frac{15}{48}\), we can divide both the numerator and the denominator by their greatest common divisor, which is 3:
\[ \frac{15 \div 3}{48 \div 3} = \frac{5}{16} \]

### Final Answer:
The probability that both disks selected are green is \(\frac{5}{16}\).

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