A box contains 2 blue caps, 4 red caps, 5 green caps and 1 yellow cap.
if two caps are picked at random, what is the probability that both are blue?

Options

A. `1/6`
B. `1/10`
C. `1/12`
D. None of the above

Answer

To find the probability that both caps picked are blue, we can follow these steps:

### Step 1: Determine the total number of caps
First, we need to find the total number of caps in the box. The box contains:
- 2 blue caps
- 4 red caps
- 5 green caps
- 1 yellow cap

**Total number of caps = 2 (blue) + 4 (red) + 5 (green) + 1 (yellow) = 12 caps.**

### Step 2: Calculate the total number of ways to pick 2 caps
Next, we need to calculate the total number of ways to choose 2 caps from the 12 caps. This can be calculated using the combination formula:

\[ \text{Total ways to pick 2 caps} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

Where \( n \) is the total number of caps (12) and \( r \) is the number of caps to pick (2).

\[ \binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66 \]

### Step 3: Calculate the number of ways to pick 2 blue caps
Now, we need to find the number of ways to choose 2 blue caps from the 2 available blue caps. Again, we use the combination formula:

\[ \text{Ways to pick 2 blue caps} = \binom{2}{2} = \frac{2!}{2!(2-2)!} = 1 \]

### Step 4: Calculate the probability
Finally, we can calculate the probability that both caps picked are blue. The probability \( P \) can be calculated as:

\[ P(\text{both caps are blue}) = \frac{\text{Number of ways to pick 2 blue caps}}{\text{Total ways to pick 2 caps}} = \frac{1}{66} \]

### Final Answer
Thus, the probability that both caps picked are blue is:

\[ \frac{1}{66} \]

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