In a bag there are 6 blue , 4 red and 5 green balls . Three balls are chosen at random with replacement ,find probability of getting utmost one color .

Options

A. `(27)/(125)`
B. `(64)/(125)`
C. `(64)/(3375)`
D. `(3)/(25)`

Answer

To solve the problem of finding the probability of getting at most one color when choosing three balls from a bag containing 6 blue, 4 red, and 5 green balls, we will follow these steps:

### Step 1: Determine the total number of balls
We first need to find the total number of balls in the bag.

- Number of blue balls = 6
- Number of red balls = 4
- Number of green balls = 5

**Total number of balls = 6 + 4 + 5 = 15**

### Step 2: Understand the requirement
We need to find the probability of getting at most one color when choosing three balls. This means we can either get:
1. All three balls of the same color.
2. Two balls of one color and one ball of another color (which would not meet the "at most one color" condition, so we will not consider this case).

### Step 3: Calculate the probability of getting all three balls of the same color
1. **All Blue Balls:**
- Probability of choosing a blue ball = Number of blue balls / Total number of balls = 6/15
- Probability of choosing three blue balls = (6/15) × (6/15) × (6/15) = (6/15)³ = 216/3375

2. **All Red Balls:**
- Probability of choosing a red ball = Number of red balls / Total number of balls = 4/15
- Probability of choosing three red balls = (4/15) × (4/15) × (4/15) = (4/15)³ = 64/3375

3. **All Green Balls:**
- Probability of choosing a green ball = Number of green balls / Total number of balls = 5/15
- Probability of choosing three green balls = (5/15) × (5/15) × (5/15) = (5/15)³ = 125/3375

### Step 4: Add the probabilities
Now, we will add the probabilities of getting all blue, all red, and all green balls:

- Total probability = Probability of all blue + Probability of all red + Probability of all green
- Total probability = (216/3375) + (64/3375) + (125/3375)

**Total probability = (216 + 64 + 125) / 3375 = 405 / 3375**

### Step 5: Simplify the probability
Now we simplify the fraction:

- 405 and 3375 can be simplified by dividing both by 15.
- 405 ÷ 15 = 27
- 3375 ÷ 15 = 225

Thus, the simplified probability is:

**Probability = 27 / 225**

### Step 6: Final simplification
We can simplify this further:

- 27 and 225 can be divided by 9.
- 27 ÷ 9 = 3
- 225 ÷ 9 = 25

Thus, the final probability is:

**Probability = 3 / 25**

### Conclusion
The probability of getting at most one color when choosing three balls from the bag is **3/25**.

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