In a lottery, there are 10 prize tickets and 25 blank tickets. One ticket is drawn at random. What is the probability that the ticket drawn is a prize ticket?

Introduction / Context:
This is a simple probability problem dealing with a finite set of equally likely outcomes. The lottery contains a certain number of prize tickets and a certain number of blank tickets. The question asks for the probability of drawing a prize ticket in a single random draw. It tests basic understanding of probability as the ratio of favourable outcomes to total possible outcomes.

Given Data / Assumptions:

Number of prize tickets = 10.

Number of blank tickets = 25.

Total tickets in the lottery = prize tickets + blank tickets.

One ticket is drawn at random, and each ticket is equally likely to be chosen.

Concept / Approach:
The theoretical probability of an event is defined as the number of favourable outcomes divided by the total number of equally likely outcomes. Here, the favourable outcomes are the prize tickets. The total outcomes are all tickets, both prize and blank. Therefore, the required probability is simply the ratio 10 divided by the total number of tickets. Simplifying this fraction gives the final answer.

Step-by-Step Solution:
Total number of tickets = number of prize tickets + number of blank tickets.Total tickets = 10 + 25 = 35.Number of favourable tickets (prize tickets) = 10.Required probability = favourable / total = 10 / 35.Simplify 10/35 by dividing numerator and denominator by 5.10/35 = 2/7.Thus, the probability of drawing a prize ticket is 2/7.
Verification / Alternative check:
As a quick check, note that 2/7 is less than 1/2, which makes sense because prizes are fewer than blanks (10 versus 25).Also, the probability of drawing a blank is 25/35 = 5/7.2/7 + 5/7 = 1, which confirms that the probabilities of all possible outcomes sum to 1.
Why Other Options Are Wrong:
5/7 is the probability of drawing a blank ticket, not a prize. 1/5 and 1/2 do not reflect the actual ratio 10/35. 3/7 would imply 15 prize tickets, which is not the case here. Only 2/7 correctly represents 10 favourable outcomes out of 35 total tickets.

Common Pitfalls:
Students sometimes mistakenly use only the number of blanks or only the number of prizes as the denominator. It is important to always include all possible outcomes in the denominator. Another error is failing to simplify the fraction; while 10/35 is correct, exams usually expect the simplest form 2/7. Always check if the fraction can be reduced.

Final Answer:
The probability that the ticket drawn is a prize ticket is 2/7.