Tickets are numbered consecutively from 101 to 350 and placed in a box. If one ticket is selected at random, what is the probability that its hundreds digit is 2?

Introduction / Context:
This problem deals with place value and probability in a simple discrete setting. Tickets are numbered from 101 to 350, and we draw one ticket at random. The question asks for the probability that the hundreds digit of the selected ticket is 2. This tests counting ranges and understanding of digit positions in numbers.

Given Data / Assumptions:

• Tickets are numbered from 101 to 350 inclusive.
• All tickets are equally likely to be selected.
• We are interested in numbers whose hundreds digit is 2.
• Hundreds digit refers to the digit in the hundreds place of a three digit number.

Concept / Approach:
We need two counts: the total number of tickets and the number of tickets whose hundreds digit is 2. Once we have both, probability is favourable tickets divided by total tickets. Numbers with hundreds digit 2 lie between 200 and 299 inclusive. We also need to confirm that this block is fully inside the overall range from 101 to 350.

Step-by-Step Solution:
Total tickets from 101 to 350 inclusive = 350 - 101 + 1. Compute 350 - 101 = 249, then add 1 to get 250 tickets in total. Numbers with hundreds digit 2 are all integers from 200 to 299 inclusive. Count of numbers from 200 to 299 inclusive = 299 - 200 + 1 = 100. All of these 100 numbers lie inside the given range 101 to 350. Therefore, favourable tickets = 100, total tickets = 250. Probability = favourable / total = 100 / 250 = 2 / 5. Convert 2 / 5 to decimal: 2 divided by 5 = 0.40.

Verification / Alternative check:
We can classify by hundreds digit: from 101 to 199 (hundreds digit 1), 200 to 299 (hundreds digit 2), and 300 to 350 (hundreds digit 3). The counts in each block are 99, 100, and 51 respectively, which sum to 250. Only the middle block has hundreds digit 2, confirming that there are exactly 100 favourable tickets. The probability 100/250 reduces to 2/5, which equals 0.40, confirming our result.

Why Other Options Are Wrong:
0.25: This would correspond to 62.5 favourable tickets, which is not an integer and does not match the actual count of 100.
0.5: This would imply that half of the tickets have hundreds digit 2, but only 100 out of 250 do.
0.75: This would mean three quarters of the tickets are favourable, which is far too high.
None of these: This is incorrect because 0.40 is listed and is the correct probability.

Common Pitfalls:
Learners sometimes forget to include both endpoints when counting inclusively, which leads to errors of plus or minus one. Another common mistake is to misinterpret the hundreds digit and instead count numbers containing digit 2 anywhere. It is important to focus only on the hundreds place. Also, mixing up decimal and fractional representations can cause confusion, so reducing to 2/5 and then converting to 0.40 is a good habit.

Final Answer:
Thus, the probability that a randomly selected ticket has hundreds digit 2 is 0.40.