GATE 2023 BT – Question 61
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 ___________.
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Correct answer: 0.44 to 0.46
Explanation
**Sample space.** The first draw is replaced, so the two draws are independent: $10\times10=100$ equally likely ordered pairs.
**By symmetry.** The pairs split into three groups:
- equal numbers: 10 pairs,
- second greater than first,
- second less than first.
The last two groups are mirror images of each other, so they have the same size:
$$\frac{100-10}{2}=45\text{ pairs each}.$$
$$P(\text{second}>\text{first})=\frac{45}{100}=\mathbf{0.45}.$$