An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn, the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is red is

Options

A. `(26)/(49)`
B. `(32)/(49)`
C. `(27)/(49)`
D. `(21)/(49)`

Answer

To solve the problem step by step, we will analyze the situation based on the two possible outcomes of the first draw and calculate the probabilities accordingly.

### Step 1: Initial Setup
We have an urn containing:
- 5 Red balls
- 2 Green balls

Total balls = 5 + 2 = 7 balls.

### Step 2: Calculate the Probability of Drawing a Green Ball
The probability of drawing a green ball first (let's denote this event as G) is given by:
\[ P(G) = \frac{\text{Number of Green Balls}}{\text{Total Balls}} = \frac{2}{7} \]

### Step 3: Update the Urn After Drawing a Green Ball
If a green ball is drawn, we add one red ball to the urn. The new composition of the urn will be:
- Red balls = 5 + 1 = 6
- Green balls = 2 - 1 = 1

Total balls now = 6 + 1 = 7.

### Step 4: Calculate the Probability of Drawing a Red Ball After Drawing a Green Ball
Now, we calculate the probability of drawing a red ball (let's denote this event as R) after having drawn a green ball:
\[ P(R | G) = \frac{\text{Number of Red Balls}}{\text{Total Balls}} = \frac{6}{7} \]

### Step 5: Calculate the Combined Probability for the First Scenario
The combined probability for the first scenario (drawing a green ball first and then a red ball) is:
\[ P(G) \cdot P(R | G) = \frac{2}{7} \cdot \frac{6}{7} = \frac{12}{49} \]

### Step 6: Calculate the Probability of Drawing a Red Ball
Now, let's consider the second scenario where a red ball is drawn first (let's denote this event as R'). The probability of drawing a red ball first is:
\[ P(R') = \frac{\text{Number of Red Balls}}{\text{Total Balls}} = \frac{5}{7} \]

### Step 7: Update the Urn After Drawing a Red Ball
If a red ball is drawn, we add one green ball to the urn. The new composition of the urn will be:
- Red balls = 5 - 1 = 4
- Green balls = 2 + 1 = 3

Total balls now = 4 + 3 = 7.

### Step 8: Calculate the Probability of Drawing a Red Ball After Drawing a Red Ball
Now, we calculate the probability of drawing a red ball after having drawn a red ball:
\[ P(R | R') = \frac{\text{Number of Red Balls}}{\text{Total Balls}} = \frac{4}{7} \]

### Step 9: Calculate the Combined Probability for the Second Scenario
The combined probability for the second scenario (drawing a red ball first and then a red ball again) is:
\[ P(R') \cdot P(R | R') = \frac{5}{7} \cdot \frac{4}{7} = \frac{20}{49} \]

### Step 10: Total Probability of Drawing a Red Ball on the Second Draw
Finally, we sum the probabilities from both scenarios to find the total probability of drawing a red ball on the second draw:
\[ P(\text{Second ball is Red}) = P(G) \cdot P(R | G) + P(R') \cdot P(R | R') = \frac{12}{49} + \frac{20}{49} = \frac{32}{49} \]

Thus, the probability that the second ball drawn is red is:
\[ \boxed{\frac{32}{49}} \]