Question

If a die is thrown at random, then the expectation of the number on it is

• A. \( 2 \cdot 4 \)
• B. \( 3 \cdot 5 \)
• C. \( 2 \cdot 1 \)
• D. \( 3 \cdot 3 \)

Hint:

To calculate the expected value of a discrete uniform distribution, sum all possible outcomes and divide by the number of outcomes.

Answer 1

The Correct Option is B

Step 1: Understand the expected value formula.
The expected value \( E(X) \) of a discrete random variable \( X \) is given by the sum of all possible outcomes weighted by their probabilities. For a die, the possible outcomes are 1, 2, 3, 4, 5, and 6, and the probability of each outcome is \( \frac{1}{6} \).
Step 2: Calculate the expected value.
The expectation of the number on the die is: \[ E(X) = \frac{1}{6} \cdot (1 + 2 + 3 + 4 + 5 + 6) = \frac{21}{6} = 3.5. \] Thus, the expected value is \( 3.5 \), which corresponds to option (B) \( 3 \cdot 5 \).
Step 3: Conclusion.
Thus, the correct answer is \( 3 \cdot 5 \), corresponding to option (B).