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

Option b: 1.50. Solution: Probability of Getting At Least 6 Heads in 8 Coin Tosses Let X be the number of heads obtained in n = 8 independent tosses of an unbiased coin. The probability of success (getting a head) is p = 1/2 , and the probability of failure is q = 1/2 . We use the Binomial Distribution formula: P(X = k) = (n choose k) pk qn-k P(X = k) = (8 choose k) (1/2)k (1/2)8-k = (8 choose k) (1/2)8 = (8/k choose 256) We want to find P(X ≥ 6) : P(X ≥ 6) = P(X = 6) + P(X = 7) + P(X = 8) Calculate the combinations: - (8 choose 6) = 8 × 7/2 = 28 - (8 choose 7) = 8 - (8 choose 8) = 1 Sum of favorable combinations = 28 + 8 + 1 = 37 . Thus, the probability is: P(X ≥ 6) = 37/256 Hence, Option B is the correct answer.

ProbabilityMCQMTP Nov 21Question 3408 of 187

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

For any discrepancies in this question, email contact@cadada.in

Correct Answer

✅ Option b — $\displaystyle 1.50$

All Options:

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

Detailed Solution & Explanation

**Probability of Getting At Least 6 Heads in 8 Coin Tosses** Let

\displaystyle X

be the number of heads obtained in

\displaystyle n = 8

independent tosses of an unbiased coin. The probability of success (getting a head) is

\displaystyle p = \frac{1}{2}

, and the probability of failure is

\displaystyle q = \frac{1}{2}

. We use the Binomial Distribution formula:

P(X = k) = \binom{n}{k} p^k q^{n-k}

P(X = k) = \binom{8}{k} \left(\frac{1}{2}\right)^k \left(\frac{1}{2}\right)^{8-k} = \binom{8}{k} \left(\frac{1}{2}\right)^8 = \frac{\binom{8}{k}}{256}

We want to find

\displaystyle P(X \ge 6)

P(X \ge 6) = P(X = 6) + P(X = 7) + P(X = 8)

Calculate the combinations: -

\displaystyle \binom{8}{6} = \frac{8 \times 7}{2} = 28

\displaystyle \binom{8}{7} = 8

\displaystyle \binom{8}{8} = 1

Sum of favorable combinations =

\displaystyle 28 + 8 + 1 = 37

. Thus, the probability is:

P(X \ge 6) = \frac{37}{256}

Hence, **Option B** 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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