Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange ma

Q: Wait, why do we multiply the probabilities? Shouldn't we add them?

That is a classic trap! We add probabilities when we are looking for the chance of event A OR event B happening (mutually exclusive events). But here, we need the first marble to be red AND the second to be white. When you need a sequence of events to occur together, you multiply their individual probabilities.

Q: What would change if the problem said 'without replacement'?

Great question! If we didn't replace the first marble, the total number of marbles for the second draw would drop from 75 to 74. Furthermore, if the first marble drawn was white, the number of white marbles available for the second draw would also change. 'With replacement' keeps the system stable and the events independent.

Q: Does the order of the draws matter?

In this specific question, yes. We are looking for a specific sequence: Red first, then White. If the question asked for 'one red and one white' without specifying the order, we would have to consider both (Red, White) and (White, Red) scenarios. But here, the order is fixed, so we just calculate that one specific path.

Question

Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white, is

Options:

(A) $\frac{2}{25}$

(B) $\frac{4}{25}$

(C) $\frac{2}{3}$

(D) $\frac{4}{75}$

Sigma Percentile · Accepted Answer

### The Dance of Probability: A Journey into Independence

Welcome, future engineer! Today, we are not just solving a probability problem; we are stepping into the shoes of a statistician, peering into a box of possibilities, and learning the fundamental language of uncertainty.

Probability is the art of predicting the future, and to master it, we must first master the art of visualization.

### The Universe in a Box

Imagine you are standing before a box. Inside this box lies a microcosm of our problem: a collection of 75 marbles. We have 10 red, 30 white, 20 blue, and 15 orange.

Before we do anything, we must define our universe. The total number of outcomes, or our sample space $n(S)$, is the sum of all these possibilities:

$$n(S) = 10 + 30 + 20 + 15 = 75$$

This number, 75, is our anchor. It is the denominator for every probability we calculate in this scenario.

### The First Step: The Red Draw

We reach into the box. We want a red marble. There are 10 red marbles out of 75. The probability of this first event, $P(R_1)$, is simply the number of favorable outcomes divided by the total outcomes:

$$P(R_1) = \frac{10}{75}$$

Before we rush to multiply, let's pause and simplify. Mathematics is about elegance, and working with smaller numbers is always safer. Dividing both the numerator and the denominator by 5, we get:

$$P(R_1) = \frac{2}{15}$$

### The Magic of 'With Replacement'

Now, here is where most students stumble. The problem states: 'with replacement.' This is the most critical piece of information.

In the world of probability, this phrase is a reset button. It means that after we observe the first marble, we put it back. The box is restored to its original state.

The second draw has no memory of the first. This is the definition of **independent events**. Because of this, the total count for the second draw remains 75, and the count of white marbles remains 30.

### The Second Step: The White Draw

Now, we reach in again, hoping for a white marble. Since we replaced the first marble, the box is exactly as it was at the start. We have 30 white marbles out of 75.

The probability of this second event, $P(W_2)$, is:

$$P(W_2) = \frac{30}{75}$$

Again, let's simplify. Both 30 and 75 are divisible by 15:

$$P(W_2) = \frac{2}{5}$$

### The Synthesis: The Multiplication Rule

We have the probability of the first event and the probability of the second. But we need the probability of both happening in succession.

This is where the multiplication rule comes in. When we need event $A$ AND event $B$ to occur, we multiply their probabilities:

$$P(R_1 \cap W_2) = P(R_1) 	imes P(W_2)$$

This is the moment of truth. We take our simplified fractions and combine them:

$$P = \frac{2}{15} 	imes \frac{2}{5}$$

Multiplying the numerators ($2 	imes 2 = 4$) and the denominators ($15 	imes 5 = 75$), we arrive at our destination:

$$P = \frac{4}{75}$$

### The Takeaway

Look at that result: $\frac{4}{75}$. It is clean, precise, and logically sound.

The beauty of this problem lies in the independence of the events. By replacing the marble, we ensured that the second draw was not 'tainted' by the first.

As you continue your JEE journey, remember this: always identify whether events are independent or dependent before you start your calculations. It is the difference between a correct answer and a common trap. Keep practicing, keep visualizing, and keep falling in love with the logic behind the math!

Animated Solution for Mathematics - Probability: Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white, is

Select Answer:

Visualized Solution

Visualizing the Box

Total Sample Space $n (S)$

First Event: Red Marble

Simplifying $P (R_{1})$

The 'With Replacement' Rule

Second Event: White Marble

Simplifying $P (W_{2})$

Multiplication Rule

Final Calculation

Conclusion & Takeaway

The Sigma Insight: Classical Definition of Probability

The Dance of Probability

The Universe in a Box

The First Step

The Magic of 'With Replacement'

The Second Step

The Synthesis

The Takeaway

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