A small bag contains 4 white and 3 red marbles . Two marbles are randmly removed from the bag . Find the probability that a white marble is removed , followed by a red.

Options

A. `1/7`
B. `2/7`
C. `3/7`
D. `4/7`

Answer

To solve the problem of finding the probability that a white marble is removed followed by a red marble from a bag containing 4 white and 3 red marbles, we can follow these steps:

### Step 1: Determine the total number of marbles.
The bag contains:
- 4 white marbles
- 3 red marbles

Total number of marbles = 4 (white) + 3 (red) = 7 marbles.

**Hint:** Always start by calculating the total number of items in the set.

### Step 2: Calculate the probability of drawing a white marble first.
The probability of drawing a white marble first is given by the ratio of the number of white marbles to the total number of marbles.

\[ P(\text{White first}) = \frac{\text{Number of white marbles}}{\text{Total number of marbles}} = \frac{4}{7} \]

**Hint:** The probability of an event is the number of favorable outcomes divided by the total number of outcomes.

### Step 3: Update the counts after drawing a white marble.
After drawing one white marble, the counts are:
- White marbles left = 4 - 1 = 3
- Red marbles left = 3 (remains unchanged)

Total marbles left = 3 (white) + 3 (red) = 6 marbles.

**Hint:** Always update the counts after each draw to reflect the new situation.

### Step 4: Calculate the probability of drawing a red marble second.
Now, we need to find the probability of drawing a red marble after having drawn a white marble.

\[ P(\text{Red second | White first}) = \frac{\text{Number of red marbles}}{\text{Total number of marbles left}} = \frac{3}{6} = \frac{1}{2} \]

**Hint:** Conditional probability is calculated based on the new total after the first event.

### Step 5: Calculate the combined probability.
To find the overall probability of both events happening (drawing a white marble first and then a red marble), we multiply the probabilities from Steps 2 and 4.

\[ P(\text{White first and Red second}) = P(\text{White first}) \times P(\text{Red second | White first}) \]

\[ = \frac{4}{7} \times \frac{1}{2} = \frac{4 \times 1}{7 \times 2} = \frac{4}{14} = \frac{2}{7} \]

**Hint:** The probability of two independent events occurring in sequence is the product of their individual probabilities.

### Final Answer:
The probability that a white marble is removed followed by a red marble is \( \frac{2}{7} \).