ProbabilityPYQ Sept 25Question 4479 of 187
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The number of tosses of a coin, that are needed so that the probability of getting at least one head is 0.875, is

Options

A2
B3
C4
D5
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Correct Answer

✅ Option b — 3

All Options:

  • A2
  • B3
  • C4
  • D5

Detailed Solution & Explanation

Let n\displaystyle n be the number of independent tosses of a fair coin. We want to find n\displaystyle n such that the probability of getting at least one head is 0.875\displaystyle 0.875.
1. **Define the probabilities of a single toss**:
- Probability of getting a head: p=12\displaystyle p = \frac{1}{2}
- Probability of getting a tail: q=12\displaystyle q = \frac{1}{2}
2. **Probability of getting no heads** in n\displaystyle n tosses:
Since each toss is independent, the probability of getting all tails is:
P(No Heads)=qn=(12)nP(\text{No Heads}) = q^n = \left(\frac{1}{2}\right)^n
3. **Probability of getting at least one head** in n\displaystyle n tosses:
P(At least one Head)=1−P(No Heads)=1−(12)nP(\text{At least one Head}) = 1 - P(\text{No Heads}) = 1 - \left(\frac{1}{2}\right)^n
4. **Solve for n\displaystyle n**:
Set the probability equal to 0.875\displaystyle 0.875:
1−(12)n=0.8751 - \left(\frac{1}{2}\right)^n = 0.875
(12)n=1−0.875\left(\frac{1}{2}\right)^n = 1 - 0.875
(12)n=0.125\left(\frac{1}{2}\right)^n = 0.125
Convert 0.125\displaystyle 0.125 to a fraction:
0.125=1251000=18=(12)30.125 = \frac{125}{1000} = \frac{1}{8} = \left(\frac{1}{2}\right)^3
Comparing the exponents:
(12)n=(12)3  ⟹  n=3\left(\frac{1}{2}\right)^n = \left(\frac{1}{2}\right)^3 \implies n = 3
Thus, the required number of tosses is 3\displaystyle 3.
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.

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