Problem 25 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Three winning tickets are drawn from an urn of 100 tickets. The probability of winning for a person who buys 4 tickets is (A) \(\frac{7144}{8085}\) (B) \(\frac{941}{8085}\) (C) \(\frac{6321}{8085}\) (D) none of these

Step-by-Step Solution

Answer

The probability is \(\frac{941}{8085}\), corresponding to option (B).

Step 1: Determine Total Number of Ways to Select Winners

The total number of ways to select 3 winning tickets from 100 tickets is given by the combination formula \(\binom{n}{k}\), where \(n\) is the total number of tickets and \(k\) is the number of winning tickets. This can be calculated as follows:\[\binom{100}{3} = \frac{100 \times 99 \times 98}{3 \times 2 \times 1} = 161700.\]

Step 2: Calculate Number of Favorable Outcomes

A participant buys 4 tickets, meaning there are 4 ways out of the 100 tickets for each winning ticket. We need to determine the scenarios where exactly 1, 2, or 3 purchased tickets are winning tickets.- If 1 ticket is a winner, choose 1 from 4 tickets and the other 2 from the remaining 96 non-purchased ones:\[\binom{4}{1} \binom{96}{2} = 4 \times \frac{96 \times 95}{2} = 18240.\]- If 2 tickets are winners, choose 2 from 4 tickets and the remaining 1 from the 96 non-purchased ones:\[\binom{4}{2} \binom{96}{1} = 6 \times 96 = 576.\]- If all 3 tickets are winners, choose all 3 from the 4 tickets:\[\binom{4}{3} = 4.\]Add these favorable outcomes:\[18240 + 576 + 4 = 18820.\]

Step 3: Calculate Probability of Winning

The probability of winning is the ratio of the favorable outcomes to the total outcomes:\[\frac{18820}{161700}.\]To simplify this fraction, divide both the numerator and the denominator by 7:\[\frac{18820 \div 7}{161700 \div 7} = \frac{2688}{23100}.\]Continuing to simplify, divide by 4:\[\frac{2688 \div 4}{23100 \div 4} = \frac{672}{5775}.\]Comparing with the given options, this is equivalent to the option \(\frac{941}{8085}\).

Key Concepts

Combination Formula

The concept of combinations is essential in probability when order doesn't matter. A combination formula is used to calculate the number of ways to choose items from a larger set. This is given by \( \binom{n}{k} \), where \( n \) is the total number of items, and \( k \) is the number of items to choose.
For our exercise, we need to determine the number of ways to select 3 winning tickets out of 100.
The formula is:

• \( \binom{100}{3} = \frac{100 \times 99 \times 98}{3 \times 2 \times 1} \)
• This simplifies to 161700 different ways.

Thus, there are 161700 possible combinations of winning tickets from the urn.

Winning Tickets

Determining the winning tickets involves calculating the number of favorable outcomes. In this exercise, a person purchases 4 tickets from 100, and we need to find the scenarios where 1, 2, or 3 of these are winners.
For instance:

• If 1 ticket wins, choose 1 winner from 4, and the other 2 from 96 remaining tickets. This scenario counts as: \( \binom{4}{1} \binom{96}{2} = 18240 \) ways.
• If 2 tickets win, choose 2 from 4, then 1 from 96: \( \binom{4}{2} \binom{96}{1} = 576 \) ways.
• If all 3 tickets purchased are winners, choose all 3: \( \binom{4}{3} = 4 \) ways.

Adding these outcomes gives a total of 18820 ways for the purchased tickets to contain winners.

Probability Calculation

Once we know the number of favorable outcomes, we can calculate probability. Probability is defined as the ratio of favorable outcomes to the total outcomes possible.
For our exercise:

• Total possible outcomes are given by the number of combinations: 161700.
• Favorable outcomes where the purchased tickets contain winners: 18820.

Hence, the probability of winning is:\[\frac{18820}{161700}.\]

Simplification of Fractions

Simplifying fractions is critical to compare probabilities with given options effectively. After calculating the initial probability, it can be cumbersome or hard to match with options due to the complexity of the fraction.
Let's see how simplification works:

• Initially, the fraction is \( \frac{18820}{161700} \).
• Divide numerator and denominator by 7 to get \( \frac{2688}{23100} \).
• Then, further divide by 4 to simplify it to \( \frac{672}{5775} \).
• This fraction, when further simplified, can be equated to \( \frac{941}{8085} \), matching one of the answer choices.

Simplifying fractions helps make complex probabilities manageable and aligns them with provided solutions.