ProbabilityMCQMTP Jun 24 Series IIQuestion 3369 of 187

In a box carrying one dozen of oranges, one third has become bad. If $\displaystyle 3$ oranges are taken out from the box at random, what is the probability that at least one orange out of the three oranges picked up is good?

Options

\displaystyle 54/55

\displaystyle 1/55

\displaystyle 43/50

DNone of these

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Correct Answer

✅ Option a — $\displaystyle 54/55$

All Options:

A $\displaystyle 54/55$
B $\displaystyle 1/55$
C $\displaystyle 43/50$
DNone of these

Detailed Solution & Explanation

**Probability of Selecting At Least One Good Orange** Total oranges = 12 (one dozen) Given: - Bad oranges =

\displaystyle \frac{1}{3} \times 12 = 4

- Good oranges =

\displaystyle 12 - 4 = 8

Three oranges are picked at random. We want to find the probability that at least one is good. Using the complement rule:

P(\text{at least one good}) = 1 - P(\text{all three bad})

**Step 1: Calculate total ways to choose 3 oranges**

\text{Total ways} = \binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220

**Step 2: Calculate ways to choose 3 bad oranges**

\text{Favorable ways for all bad} = \binom{4}{3} = 4

**Step 3: Calculate the probability**

P(\text{all three bad}) = \frac{4}{220} = \frac{1}{55}

P(\text{at least one good}) = 1 - \frac{1}{55} = \frac{54}{55}

Hence, **Option A** 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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In a box carrying one dozen of oranges, one third has become bad. If 3\displaystyle 33 oranges are taken out from the box at random, what is the probability that at least one orange out of the three oranges picked up is good?