A bag contains three white, two black and four red balls. If four balls are drawn at random with replacement, the probability that the sample contains just one white ball is

Options

A. `16/81`
B. `8/81`
C. `32/81`
D. `4/81`

Answer

To solve the problem of finding the probability that exactly one white ball is drawn when four balls are drawn at random with replacement from a bag containing three white, two black, and four red balls, we can follow these steps:

### Step 1: Determine the total number of balls
The bag contains:
- 3 white balls
- 2 black balls
- 4 red balls

Total number of balls = 3 + 2 + 4 = 9 balls.

### Step 2: Calculate the probability of drawing a white ball
The probability of drawing a white ball (P(W)) is given by the ratio of the number of white balls to the total number of balls:
\[ P(W) = \frac{3}{9} = \frac{1}{3} \]

### Step 3: Calculate the probability of drawing a non-white ball
The probability of drawing a non-white ball (P(NW)) is the complement of drawing a white ball:
\[ P(NW) = 1 - P(W) = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 4: Use the binomial probability formula
We want to find the probability of drawing exactly one white ball in four draws. This follows a binomial distribution where:
- n = number of trials (draws) = 4
- k = number of successes (drawing a white ball) = 1
- p = probability of success (drawing a white ball) = \( \frac{1}{3} \)
- q = probability of failure (drawing a non-white ball) = \( \frac{2}{3} \)

The formula for the probability of exactly k successes in n trials is given by:
\[ P(X = k) = \binom{n}{k} p^k q^{n-k} \]

### Step 5: Substitute the values into the formula
Substituting the values we have:
\[ P(X = 1) = \binom{4}{1} \left(\frac{1}{3}\right)^1 \left(\frac{2}{3}\right)^{4-1} \]

Calculating the binomial coefficient:
\[ \binom{4}{1} = 4 \]

Now substituting this into the equation:
\[ P(X = 1) = 4 \left(\frac{1}{3}\right)^1 \left(\frac{2}{3}\right)^3 \]
\[ = 4 \cdot \frac{1}{3} \cdot \left(\frac{2^3}{3^3}\right) \]
\[ = 4 \cdot \frac{1}{3} \cdot \frac{8}{27} \]
\[ = 4 \cdot \frac{8}{81} \]
\[ = \frac{32}{81} \]

### Final Answer
The probability that the sample contains exactly one white ball is \( \frac{32}{81} \).

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