An urn contains 4 green , 5 blue ,2 red and 3yellow marbles .
If three marbles are drawn at random , what is the probability that at least one is yellow ?

Options

A. `(1)/(3)`
B. `(199)/(364)`
C. `(165)/(364)`
D. `(3)/(11)`

Explanation

To solve the problem of finding the probability that at least one marble drawn from the urn is yellow, we will follow these steps:

### Step 1: Determine the total number of marbles
We have:
- 4 green marbles
- 5 blue marbles
- 2 red marbles
- 3 yellow marbles

**Total number of marbles = 4 + 5 + 2 + 3 = 14**

### Step 2: Calculate the total number of ways to draw 3 marbles from 14
We need to find the number of combinations of drawing 3 marbles from 14. This can be calculated using the combination formula:

\[ C(n, r) = \frac{n!}{r!(n - r)!} \]

Where \( n \) is the total number of items, and \( r \) is the number of items to choose.

So, we calculate:

\[ C(14, 3) = \frac{14!}{3!(14 - 3)!} = \frac{14!}{3! \cdot 11!} \]

Calculating this gives:

\[ C(14, 3) = \frac{14 \times 13 \times 12}{3 \times 2 \times 1} = \frac{2184}{6} = 364 \]

### Step 3: Calculate the number of ways to draw 3 marbles with none being yellow
If none of the marbles drawn are yellow, we are only considering the green, blue, and red marbles. The total number of these marbles is:

\[ 4 \text{ (green)} + 5 \text{ (blue)} + 2 \text{ (red)} = 11 \]

Now, we calculate the number of ways to choose 3 marbles from these 11:

\[ C(11, 3) = \frac{11!}{3!(11 - 3)!} = \frac{11!}{3! \cdot 8!} \]

Calculating this gives:

\[ C(11, 3) = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = \frac{990}{6} = 165 \]

### Step 4: Calculate the probability of drawing at least one yellow marble
The probability of drawing at least one yellow marble can be found by subtracting the probability of drawing no yellow marbles from 1.

The probability of drawing no yellow marbles is given by:

\[ P(\text{none yellow}) = \frac{\text{Number of ways to choose 3 non-yellow marbles}}{\text{Total ways to choose 3 marbles}} = \frac{165}{364} \]

Thus, the probability of drawing at least one yellow marble is:

\[ P(\text{at least one yellow}) = 1 - P(\text{none yellow}) = 1 - \frac{165}{364} = \frac{364 - 165}{364} = \frac{199}{364} \]

### Final Answer
The probability that at least one marble drawn is yellow is:

\[ \frac{199}{364} \]

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