During a school sports meet, a volunteer coordinates a lucky draw game to raise funds. A bag contains 50 tokens numbered from 1 to 50. Participants pay a small fee to draw a single token at random. The organizers offer three tiers of prizes depending on the property of the drawn number. Prime numbers win a small souvenir, perfect squares win a sports kit, and multiples of 11 win a bumper prizebicycle. The volunteer carefully keeps track of every drawn token, ensuring no replacement occurs during a single round. This engaging setup helps students practically understand how event constraints and sample spaces directly affect their likelihood of winning. Observe the pattern and answer the questions below: (i) Find the probability of a participant winning a bumper prize bicycle on the 1 very first draw. (ii) Find the probability that the first drawn token contains a number which is 1 a multiple of 5. (iii) Find the probability that the first drawn token wins either a small souvenir 2 or a sports kit. OR If the first participant draws token number 25 and takes it away, find the probability that the next participant draws a token that is a perfect square number.

Answer: (i) $\frac{2}{25}$ , (ii) $\frac{1}{5}$ , (iii) $\frac{11}{25}$ OR (iii Alternative) $\frac{6}{49}$

Step-by-step solution

Step 1: Define Sample Space and Probability of Bumper Prize (i)

The total number of possible outcomes on the first draw is $50$ . A bumper prize is won if the token is a multiple of 11. The multiples of 11 between 1 and 50 are 11, 22, 33, and 44, giving 4 favorable outcomes. Thus, the probability is $4/50 = 2/25$ .

Step 2: Find Probability of Drawing a Multiple of 5 (ii)

The numbers divisible by 5 from 1 to 50 are 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50. There are 10 favorable outcomes out of 50 total tokens. The probability is therefore $10/50 = 1/5$ .

Step 3: Find Probability of Souvenir or Sports Kit (iii)

Prime numbers between 1 and 50 give 15 favorable outcomes, and perfect squares give 7 favorable outcomes. Since no prime number can be a perfect square, the sets are mutually disjoint ( $A \cap B = \emptyset$ ). The total favorable outcomes are $15 + 7 = 22$ , so the probability is $22/50 = 11/25$ .

Step 4: Evaluate Alternative Part (iii OR)

When token 25 is drawn and removed, only 49 tokens remain in the bag. Since 25 is a perfect square, the number of available perfect squares decreases from 7 to 6. Thus, the probability that the next participant draws a perfect square is $6/49$ .

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