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: Conceptual Understanding:
The "mean" of outcomes from a random event, such as rolling a die, signifies the expected value. The expected value is a probability-weighted average of all possible results.Step 2: Fundamental Formula:
The mean or expected value E[X] for a discrete random variable X is calculated using the formula:\[ E[X] = \sum_{i} x_i P(X=x_i) \]Here, \(x_i\) represents each possible outcome, and \(P(X=x_i)\) denotes its corresponding probability.Step 3: In-Depth Calculation:
First, determine the probability of each outcome. A die has 6 faces.- Outcome 1 appears on 3 faces: \(P(X=1) = \frac{3}{6} = \frac{1}{2}\).- Outcome 2 appears on 2 faces: \(P(X=2) = \frac{2}{6} = \frac{1}{3}\).- Outcome 5 appears on 1 face: \(P(X=5) = \frac{1}{6}\).(Verification: The sum of probabilities is 1: \(\frac{3}{6} + \frac{2}{6} + \frac{1}{6} = \frac{6}{6} = 1\)).Next, apply the expected value formula:\[ E[X] = (1 \times P(X=1)) + (2 \times P(X=2)) + (5 \times P(X=5)) \]\[ E[X] = \left(1 \times \frac{3}{6}\right) + \left(2 \times \frac{2}{6}\right) + \left(5 \times \frac{1}{6}\right) \]\[ E[X] = \frac{3}{6} + \frac{4}{6} + \frac{5}{6} \]\[ E[X] = \frac{3+4+5}{6} = \frac{12}{6} = 2 \]Step 4: Conclusive Result:
The mean value resulting from rolling the die is 2.