Question

What is the mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on 1 face?

• A. 2
• B. 4
• C. 1
• D. 8

Hint:

To calculate the mean of a discrete distribution, think of it as a weighted average. Multiply each possible value by its probability and sum the results. This is a fundamental concept in probability and statistics.

Answer 1

The Correct Option is A

Step 1: Concept Definition:
The "mean" for a random event, such as rolling a die, is equivalent to the expected value. This expected value represents a weighted average of all potential outcomes, with each outcome's weight determined by its probability.
Step 2: Calculation Method:
The mean or expected value E[X] for a discrete random variable X is computed using the formula:\[ E[X] = \sum_{i} x_i P(X=x_i) \]In this formula, \(x_i\) denotes each possible outcome, and \(P(X=x_i)\) represents the probability of that specific outcome.
Step 3: Detailed Calculation:
Initially, the probability for each outcome must be established. A standard die has 6 faces.
- The outcome '1' occurs on 3 faces, thus \(P(X=1) = \frac{3}{6} = \frac{1}{2}\).
- The outcome '2' occurs on 2 faces, thus \(P(X=2) = \frac{2}{6} = \frac{1}{3}\).
- The outcome '5' occurs on 1 face, thus \(P(X=5) = \frac{1}{6}\).
(Verification: The sum of probabilities is 1: \(\frac{3}{6} + \frac{2}{6} + \frac{1}{6} = \frac{6}{6} = 1\)).
Subsequently, the formula for the expected value is applied:\[ E[X] = (1 \times P(X=1)) + (2 \times P(X=2)) + (5 \times P(X=5)) \]\[ E[X] = \left(1 \times \frac{3}{6}\right) + \left(2 \times \frac{2}{6}\right) + \left(5 \times \frac{1}{6}\right) \]\[ E[X] = \frac{3}{6} + \frac{4}{6} + \frac{5}{6} \]\[ E[X] = \frac{3+4+5}{6} = \frac{12}{6} = 2 \]Step 4: Final Result:
The calculated mean for the numbers obtained from rolling the die is 2.