Problem 15 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Five horses are in a race. Mr. A selects two of the horses at random and bets on them. The probability that Mr. A selected the winning horse is [2003] (a) \(\frac{2}{5}\) (b) \(\frac{4}{5}\) (c) \(\frac{3}{5}\) (d) \(\frac{1}{5}\)

Step-by-Step Solution

Answer

The probability is \(\frac{2}{5}\).

Step 1: Understand the Problem Context

Mr. A selects two horses at random from a pool of five horses and bets on these two horses.

Step 2: Calculate Total Ways to Select Horses

The total number of ways for Mr. A to select any two horses from five is given by the combination formula for choosing 2 out of 5: \( \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \).

Step 3: Calculate Ways to Select the Winning Horse

Once the winning horse has been identified, Mr. A needs to select this winning horse from the total selected horses. There are \( \binom{4}{1} \) ways to select one other horse from the remaining four. This could be combined with the winning horse, so there is just 1 way to select that specific winning horse and any one of the others. This makes \( 4 \).

Step 4: Calculate Probability of Selecting Winning Horse

The probability that Mr. A selects the winning horse is the number of favorable outcomes divided by the total number of outcomes. Thus, the probability is given by \( \frac{4}{10} = \frac{2}{5} \).

Step 5: Conclusion and Answer

After calculating, we see that Mr. A's probability of selecting the winning horse is \( \frac{2}{5} \). Thus, the correct answer is option (a).

Key Concepts

Combination Formula

When dealing with probability, especially in situations where you're deciding between groups or sets, the concept of combinations is essential. Combinations are a way of determining how many ways you can choose a subset of items from a larger set, without regard to the order of selection. To calculate the number of combinations for choosing two items from a set of five, we use the combination formula:

• The formula is expressed as \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) where \( n \) is the total number of items to choose from, and \( k \) is the number of items to choose.
• In the given problem, \( n = 5 \) and \( k = 2 \), so it becomes \( \binom{5}{2} = \frac{5!}{2!(5-2)!} = 10 \).
• Here, the factorial \( 5! \) is the product of all positive integers up to 5, and so forth for other factorials.

With this, we determine that there are 10 different ways Mr. A can choose any two horses from five for his bets.

Selection of Outcomes

Selecting outcomes from a larger group means identifying how many favorable conditions exist out of all possible situations. In the context of the horse race exercise, Mr. A needs to pick one winning horse to ensure his bet is correct.

• Since there is only one winner in every race, once the winning horse is known, any selection that includes this horse counts as a favorable outcome.
• After selecting the winning horse, Mr. A can choose any one of the remaining horses from the pool, achieved by \( \binom{4}{1} \) which equals 4. This represents the four sets including the winning horse paired with one of the other horses.

This breakdown helps to pinpoint precisely how many ways Mr. A can bet correctly if one horse wins the race.

Probability Calculation

Probability serves as a tool to gauge how likely an event is to occur. The fundamental probability concept is calculating the number of favorable outcomes divided by the total number of possible outcomes.

• In our horse race problem, the total number of outcomes is the combinations of any two horses from five, which was calculated as 10 using \( \binom{5}{2} \).
• The number of favorable outcomes, where Mr. A's choice includes the winning horse, was found to be 4 as identified earlier.
• Thus, the probability that Mr. A selects the winning horse is \( \frac{4}{10} \), which simplifies to \( \frac{2}{5} \).

This simple division gives us a clear insight into how often Mr. A can expect to win with his chosen strategy.

Mathematical Problem Solving

Mathematical problem-solving is the ability to use various mathematical concepts and tools to tackle real-life problems. With the horse race exercise, we engage in this process through several crucial steps:

• First, identify what is being asked—determining the probability of selecting a winning horse out of a group.
• Next, break down the problem into simpler, manageable parts. Calculate the total number of combinations for selecting any two horses. Then find the combinations that include the winning horse.
• Connect these calculations to produce a probability result, leveraging combination and probability formulas effectively.

This exercise emphasizes structured thinking and careful step-by-step calculations, two essential attributes in mathematical problem-solving.