There are five bags each containing identical sets of ten distinct chocolates.
One chocolate is picked from each bag.
The probability that at least two chocolates are identical is
Options:
(A) Q.3024
(B) Q.4235
(C) Q.6976
(D) Q.8125
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2021 Q.8)**
**1. Governing Concept**:
- **Topic**: Probability Theory & Complementary Probability
- Total bags $= 5$, each bag contains an identical set of $10$ distinct chocolates.
- One chocolate is picked uniformly at random from each of the 5 bags.
- Total possible outcomes:
$n(S) = 10^5 = 100,000$
- Complement event $E'$: All 5 chosen chocolates are distinct.
- Number of ways to choose 5 distinct chocolates out of 10 without replacement across bags:
$n(E') = 10 \times 9 \times 8 \times 7 \times 6 = 30,240$
- Probability that all 5 chocolates are distinct:
$P(E') = \frac{30,240}{100,000} = 0.3024$
- Probability that at least two chocolates are identical:
$P(\text{at least two identical}) = 1 - P(E') = 1 - 0.3024 = 0.6976$
**2. Option Analysis**:
- **Option (A) [0.3024] is INCORRECT**: Represents $P(\text{all distinct})$, the complementary probability.
- **Option (B) [0.4235] is INCORRECT**: Miscalculated probability.
- **Option (C) [0.6976] is CORRECT**: Exactly $1 - 0.3024 = 0.6976$.
- **Option (D) [0.8125] is INCORRECT**: Distractor.
**3. Final Answer**:
- **Correct Choice**: **C**
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