There are 5 white and 4 red balls in a bag. Three balls are drawn one by one without relacement. Find the probability distribution for the number of white balls drawn.

Explanation

To find the probability distribution for the number of white balls drawn from a bag containing 5 white and 4 red balls, we will denote the random variable \(X\) as the number of white balls drawn. The possible values for \(X\) are 0, 1, 2, and 3. We will calculate the probabilities for each of these cases.

### Step 1: Calculate Total Number of Balls
First, we note that there are a total of \(5 + 4 = 9\) balls in the bag.

### Step 2: Calculate Probability of Drawing 0 White Balls
To find the probability of drawing 0 white balls (i.e., all red balls), we can use the combination formula:

\[ P(X = 0) = \frac{\text{Number of ways to choose 3 red balls}}{\text{Total ways to choose 3 balls}} \]

Calculating the combinations:
- The number of ways to choose 3 red balls from 4 is \( \binom{4}{3} = 4 \).
- The total number of ways to choose 3 balls from 9 is \( \binom{9}{3} = 84 \).

Thus,

\[ P(X = 0) = \frac{4}{84} = \frac{1}{21} \]

### Step 3: Calculate Probability of Drawing 1 White Ball
Next, we calculate the probability of drawing 1 white ball:

\[ P(X = 1) = \frac{\text{Number of ways to choose 1 white ball and 2 red balls}}{\text{Total ways to choose 3 balls}} \]

Calculating the combinations:
- The number of ways to choose 1 white ball from 5 is \( \binom{5}{1} = 5 \).
- The number of ways to choose 2 red balls from 4 is \( \binom{4}{2} = 6 \).

Thus,

\[ P(X = 1) = \frac{5 \times 6}{84} = \frac{30}{84} = \frac{15}{42} \]

### Step 4: Calculate Probability of Drawing 2 White Balls
Now, we calculate the probability of drawing 2 white balls:

\[ P(X = 2) = \frac{\text{Number of ways to choose 2 white balls and 1 red ball}}{\text{Total ways to choose 3 balls}} \]

Calculating the combinations:
- The number of ways to choose 2 white balls from 5 is \( \binom{5}{2} = 10 \).
- The number of ways to choose 1 red ball from 4 is \( \binom{4}{1} = 4 \).

Thus,

\[ P(X = 2) = \frac{10 \times 4}{84} = \frac{40}{84} = \frac{20}{42} \]

### Step 5: Calculate Probability of Drawing 3 White Balls
Finally, we calculate the probability of drawing 3 white balls:

\[ P(X = 3) = \frac{\text{Number of ways to choose 3 white balls}}{\text{Total ways to choose 3 balls}} \]

Calculating the combinations:
- The number of ways to choose 3 white balls from 5 is \( \binom{5}{3} = 10 \).

Thus,

\[ P(X = 3) = \frac{10}{84} = \frac{5}{42} \]

### Step 6: Summarize the Probability Distribution
Now we can summarize the probability distribution:

\[ \begin{array}{|c|c|} \hline X & P(X) \\ \hline 0 & \frac{1}{21} \\ 1 & \frac{15}{42} \\ 2 & \frac{20}{42} \\ 3 & \frac{5}{42} \\ \hline \end{array} \]