Three faces of a fair die are yellow, two faces red and one blue. The die is tossed three times. The

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

Great question! We multiply because these are independent events. When you want event A AND event B AND event C to happen in a specific sequence, the multiplication rule for independent events, $P(A \cap B \cap C) = P(A) \times P(B) \times P(C)$, is the correct tool. Addition is for mutually exclusive events (OR), not for sequences (AND).

Q: What if the question didn't specify the order? Would the answer change?

Absolutely! If the question asked for one yellow, one red, and one blue in *any* order, we would have to account for all the different ways to arrange those three colors. That would be $3!$ (3 factorial) arrangements, so we would multiply our result by $6$.

Q: Why is the total number of faces 6? Is it always 6 for a die?

In this problem, yes, because the die is described as having 3 yellow, 2 red, and 1 blue faces. $3 + 2 + 1 = 6$. Always check the problem statement for the total count, as some custom dice might have different numbers of faces!

Question

Three faces of a fair die are yellow, two faces red and one blue. The die is tossed three times. The probability that the colours, yellow, red and blue, appear in the first, second and the third tosses respectively is .........

Sigma Percentile · Accepted Answer

### The Philosophy of Chance

Probability is not just a branch of mathematics; it is the elegant language of uncertainty. When we look at a problem like this, we are not just calculating numbers; we are mapping out the architecture of possibility.

Imagine you are holding this custom-painted die in your hand. It is a unique object, a microcosm of chance with $3$ yellow faces, $2$ red faces, and $1$ blue face. Our goal is to predict a specific future: a sequence of Yellow, then Red, then Blue.

### The Anatomy of the Die

Before we can predict the future, we must understand the present. The first step in any probability problem is defining the sample space.

We have $3$ yellow faces, $2$ red faces, and $1$ blue face. When you toss this die, the total number of possible outcomes for a single toss is the sum of all faces: $3 + 2 + 1 = 6$.

This number, $6$, is the denominator for our individual probabilities. It represents the entire universe of what *could* happen in one toss.

### The Logic of Independence

Now, we toss the die three times. Here is the crucial realization: the die has no memory. The result of the first toss does not change the die for the second toss.

This is the definition of **Independent Events**. Because the events are independent, the probability of the entire sequence occurring is simply the product of the individual probabilities.

We are looking for the intersection of three events:
$$P(Y \cap R \cap B) = P(Y) 	imes P(R) 	imes P(B)$$

### The Calculation

Let us break down the probability of each step:

1. **First Toss (Yellow):** We have $3$ favorable outcomes out of $6$. Thus, $P(Y) = \frac{3}{6} = \frac{1}{2}$.

2. **Second Toss (Red):** We have $2$ favorable outcomes out of $6$. Thus, $P(R) = \frac{2}{6} = \frac{1}{3}$.

3. **Third Toss (Blue):** We have $1$ favorable outcome out of $6$. Thus, $P(B) = \frac{1}{6}$.

Now, we bring them together. The probability of this specific sequence is the product of these three fractions:

$$	ext{Prob} = \frac{1}{2} 	imes \frac{1}{3} 	imes \frac{1}{6}$$

Multiplying the numerators, we get $1 	imes 1 	imes 1 = 1$. Multiplying the denominators, we get $2 	imes 3 	imes 6 = 36$.

The final probability is **$\frac{1}{36}$**.

### The "What If" Scenario

It is vital to remember that this result is specific to the order $Y ightarrow R ightarrow B$. If the question had asked for the probability of getting one of each color in *any* order, the problem would become a combinatorics challenge.

We would have to multiply our result by $3!$ (which is $6$) because there are $6$ different sequences that satisfy the condition (YRB, YBR, RYB, RBY, BYR, BRY). But here, the universe demands a specific order, and we have found it.

### Final Thoughts

Probability is about discipline. It is about identifying the sample space, recognizing independence, and executing the multiplication with precision.

You have mastered the logic of this sequence. Keep this clarity, and you will find that even the most complex probability problems are just stories waiting to be told.

Animated Solution for Mathematics - Probability: Three faces of a fair die are yellow, two faces red and one blue. The die is tossed three times. The probability that the colours, yellow, red and blue, appear in the first, second and the third tosses respectively is .........

Visualized Solution

Problem Setup: Colored Die

Total Possible Outcomes

The Target Sequence

Probability of Yellow (First Toss)

Probability of Red (Second Toss)

Probability of Blue (Third Toss)

Concept of Independent Events

Applying the Multiplication Rule

Final Calculation

Key Takeaway: Fixed Sequence

The Sigma Insight: Addition and Multiplication Theorems

The Philosophy of Chance

The Anatomy of the Die

The Logic of Independence

The Calculation

The "What If" Scenario

Final Thoughts

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