From the following probability distribution table, find E(x) x | 1 | 2 | 3f(x): | 1/2 | 1/3 | 1/6

Option c: 1.67. Solution: The expected value E(X) of the discrete random variable X is given by: E(X) = sum xi f(xi) . Using the given values: - x1 = 1 , f(x1) = 1/2 - x2 = 2 , f(x2) = 1/3 - x3 = 3 , f(x3) = 1/6 E(X) = (1 × 1/2) + (2 × 1/3) + (3 × 1/6) E(X) = 1/2 + 2/3 + 3/6 Convert to a common denominator of 6: E(X) = 3/6 + 4/6 + 3/6 = 10/6 = 5/3 ≈ 1.6667 . Rounding to two decimal places, we get 1.67 . Hence, Option C is the correct answer.

ProbabilityMCQMTP Dec 2023 Series IQuestion 2845 of 295

All Questions

From the following probability distribution table, find $\displaystyle E(x)$ x | 1 | 2 | 3f(x): | $\displaystyle \frac{1}{2}$ | $\displaystyle \frac{1}{3}$ | $\displaystyle \frac{1}{6}$

Options

\displaystyle 1

\displaystyle 1.50

\displaystyle 1.67

DNone of these

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Correct Answer

✅ Option c — $\displaystyle 1.67$

All Options:

A $\displaystyle 1$
B $\displaystyle 1.50$
C $\displaystyle 1.67$
DNone of these

Detailed Solution & Explanation

The expected value

\displaystyle E(X)

of the discrete random variable

\displaystyle X

is given by:

\displaystyle E(X) = \sum x_i f(x_i)

. Using the given values: -

\displaystyle x_1 = 1

\displaystyle f(x_1) = \frac{1}{2}

\displaystyle x_2 = 2

\displaystyle f(x_2) = \frac{1}{3}

\displaystyle x_3 = 3

\displaystyle f(x_3) = \frac{1}{6}

\displaystyle E(X) = \left(1 \times \frac{1}{2}\right) + \left(2 \times \frac{1}{3}\right) + \left(3 \times \frac{1}{6}\right)

\displaystyle E(X) = \frac{1}{2} + \frac{2}{3} + \frac{3}{6}

Convert to a common denominator of 6:

\displaystyle E(X) = \frac{3}{6} + \frac{4}{6} + \frac{3}{6} = \frac{10}{6} = \frac{5}{3} \approx 1.6667

. Rounding to two decimal places, we get

\displaystyle 1.67

. Hence, **Option C** is the correct answer.

About This Chapter: Probability

Paper

Paper 3: Quantitative Aptitude

Weightage

5-7 Marks

Key Topics

Probability Operations, Expected Value

A logic-heavy chapter dealing with random experiments, events (mutually exclusive, exhaustive), set theory probability, conditional probability, and Bayes' Theorem. It forms the basis for Theoretical Distributions.

View Official ICAI Syllabus

Exam Strategy Tip

Always draw a quick Venn Diagram or tree when faced with 'At least one' or 'Only A but not B' wording. It saves you from double-counting.

Key Concepts to Understand

Probability

A numerical measure of the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). Formula: P(A) = Number of favorable outcomes / Total number of equally likely outcomes.

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From the following probability distribution table, find E(x)\displaystyle E(x)E(x)x | 1 | 2 | 3f(x): | 12\displaystyle \frac{1}{2}21​ | 13\displaystyle \frac{1}{3}31​ | 16\displaystyle \frac{1}{6}61​