Subtopic: Fundamentals of Biomolecules, enzyme kinetics
Question Statement:
Ten playing cards numbered 1, 2, 3, …., 10 are placed face down on a table. One
card is drawn at random, its number recorded, and then replaced face down. A
card is drawn again at random. The probability that the number on the second
draw is greater than the number on the first draw (rounded off to two decimal
places) is ___________.
Official Answer Key:
0.45 to 0.45
Step-by-Step Solution & Explanation
### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2023 Q.61)**
**1. Governing Concept & Definition**:
- **Topic**: Probability & Independent Sampling with Replacement
- Cards are numbered $1, 2, \dots, 10$. Two independent draws with replacement yield ordered pairs $(x_1, x_2)$ from a total sample space of:
$N(S) = 10 \times 10 = 100\text{ equally likely outcomes}$
**2. Detailed Step-by-Step Calculation**:
- **Target Event**: The second draw is strictly greater than the first draw ($x_2 > x_1$).
- **Partitioning Sample Space**:
By symmetry:
- Outcomes where $x_1 = x_2$: Exactly 10 outcomes $(1,1), (2,2), \dots, (10,10)$.
- Outcomes where $x_1 \neq x_2$: $100 - 10 = 90$ outcomes.
- Due to identical distributions, $P(x_2 > x_1) = P(x_1 > x_2)$.
- **Calculation**:
$\text{Number of favorable pairs} = \frac{90}{2} = 45$
$P(x_2 > x_1) = \frac{45}{100} = 0.45$
**3. Final Answer**:
- **Calculated Value**: **0.45**
- **Official GATE Accepted Range**: **0.45 to 0.45**
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