Question

A die is rolled thrice. What is the probability of getting a number greater than $4$ in the first and second throws and a number less than $4$ in the third throw?

• A. $\frac{1}{3}$
• B. $\frac{1}{6}$
• C. $\frac{1}{9}$
• D. $\frac{1}{18}$

Hint:

When calculating the probability of multiple independent events, you can multiply the individual probabilities together. In this case, the events are independent (each die roll does not affect the others), so the total probability is simply the product of the individual probabilities. Make sure to simplify fractions to their lowest terms to avoid errors in final calculations.

Answer 1

The Correct Option is D

When a die is rolled, the outcomes are 1, 2, 3, 4, 5, 6. The probabilities for the given events are as follows:

A number greater than 4 includes {5, 6}. The probability of this event is:

\( P(\text{greater than 4}) = \frac{2}{6} = \frac{1}{3} \)

A number less than 4 includes {1, 2, 3}. The probability of this event is:

\( P(\text{less than 4}) = \frac{3}{6} = \frac{1}{2} \)

The probability of the required outcome (a number greater than 4 on the first and second throws, and a number less than 4 on the third throw) is the product of the probabilities:

\( P(\text{required outcome}) = P(\text{greater than 4}) \cdot P(\text{greater than 4}) \cdot P(\text{less than 4}) \)

\( P(\text{required outcome}) = \frac{1}{3} \times \frac{1}{3} \times \frac{1}{2} \)

\( P(\text{required outcome}) = \frac{1}{18} \)

Thus, the probability of the required outcome is \( \frac{1}{18} \).

Answer 2

When a die is rolled, the outcomes are 1, 2, 3, 4, 5, 6. The probabilities for the given events are as follows:

Step 1: Probability of a number greater than 4:

A number greater than 4 includes {5, 6}. The probability of this event is:

\[ P(\text{greater than 4}) = \frac{2}{6} = \frac{1}{3} \]

Step 2: Probability of a number less than 4:

A number less than 4 includes {1, 2, 3}. The probability of this event is:

\[ P(\text{less than 4}) = \frac{3}{6} = \frac{1}{2} \]

Step 3: Calculate the probability of the required outcome:

The required outcome is getting a number greater than 4 on the first and second throws, and a number less than 4 on the third throw. The probability of this outcome is the product of the individual probabilities:

\[ P(\text{required outcome}) = P(\text{greater than 4}) \cdot P(\text{greater than 4}) \cdot P(\text{less than 4}) \]

Step 4: Multiply the probabilities:

\[ P(\text{required outcome}) = \frac{1}{3} \times \frac{1}{3} \times \frac{1}{2} \]

Step 5: Final calculation:

\[ P(\text{required outcome}) = \frac{1}{18} \]

Conclusion: Thus, the probability of the required outcome is \( \frac{1}{18} \).