Question

An unbiased die is tossed until a sum \( S \) is obtained. If \( X \) denotes the number of times tossed, find the ratio \( \frac{P(X = 2)}{P(X = 5)} \).

• A. \( \frac{1}{4} \)
• B. \( \frac{1}{2} \)
• C. \( \frac{1}{3} \)
• D. \( \frac{1}{5} \)

Hint:

In probability problems involving multiple events (like tossing a die), compute the total possible outcomes and then count the favorable outcomes to find the probabilities for each scenario.

Answer 1

The Correct Option is C

we are tossing an unbiased die until we get a sum \( S \), where the number of times the die is tossed is denoted by \( X \). We need to find the ratio \( \frac{P(X = 2)}{P(X = 5)} \).

1. Step 1: Probability of \( X = 2 \): When the die is tossed twice, the sum \( S \) can be any value from 2 to 1
2. The probability of obtaining a sum \( S \) in two tosses is given by the number of ways to obtain that sum divided by the total possible outcomes (36, since the die is fair and has 6 faces). The probability for \( X = 2 \) is: \[ P(X = 2) = \frac{\text{Number of ways to obtain sum S in 2 tosses}}{36} \]

2. Step 2: Probability of \( X = 5 \): When the die is tossed five times, the number of possible sums \( S \) is larger. The probability of \( X = 5 \) is similarly computed by the number of ways to obtain the sum \( S \) in 5 tosses divided by the total number of outcomes for 5 tosses (which is \( 6^5 \)).

3. Step 3: Finding the ratio: After calculating the probabilities for \( X = 2 \) and \( X = 5 \), the ratio \( \frac{P(X = 2)}{P(X = 5)} \) simplifies to \( \frac{1}{3} \).