Problem 59 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing is: [2013] (a) \(\frac{17}{3^{5}}\) (b) \(\frac{13}{3^{5}}\) (c) \(\frac{11}{3^{5}}\) (d) \(\frac{10}{3^{5}}\)

Step-by-Step Solution

Answer

The probability that a student will get 4 or more correct by guessing is \( \frac{11}{3^5} \), option (c).

Step 1: Define the Problem Using Binomial Distribution

The problem involves finding the probability that a student will get 4 or more correct answers by guessing on a multiple-choice exam. This is a binomial distribution problem with parameters: \( n = 5 \) (number of questions) and \( p = \frac{1}{3} \) (probability of guessing a correct answer).

Step 2: Probability of Getting 4 Correct Answers

Calculate the probability of getting exactly 4 correct answers using the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Substitute \( n = 5, k = 4, p = \frac{1}{3} \) into the formula:\[ P(X = 4) = \binom{5}{4} \left(\frac{1}{3}\right)^4 \left(\frac{2}{3}\right)^{1} = 5 \times \frac{1}{81} \times \frac{2}{3} = \frac{10}{243} \]

Step 3: Probability of Getting 5 Correct Answers

Calculate the probability of getting exactly 5 correct answers. Substitute \( n = 5, k = 5, p = \frac{1}{3} \) into the formula:\[ P(X = 5) = \binom{5}{5} \left(\frac{1}{3}\right)^5 \left(\frac{2}{3}\right)^{0} = 1 \times \frac{1}{243} \times 1 = \frac{1}{243} \]

Step 4: Calculate Probability of 4 or More Correct Answers

The total probability of getting 4 or more correct answers is the sum of probabilities of getting exactly 4 correct answers and exactly 5 correct answers:\[ P(X \geq 4) = P(X = 4) + P(X = 5) = \frac{10}{243} + \frac{1}{243} = \frac{11}{243} \]

Step 5: Compare with Given Options

Express \( \frac{11}{243} \) as an equivalent fraction with a denominator of \( 3^5 \) and compare it to the options:Since \( 3^5 = 243 \), \( \frac{11}{243} = \frac{11}{3^5} \).Thus, the correct answer is option (c) \( \frac{11}{3^5} \).

Key Concepts

Understanding Probability

Probability is a fundamental concept in mathematics and statistics. It measures the likelihood that a certain event will occur. In our problem, probability helps us evaluate the chances of guessing the correct answers on a multiple-choice exam. It's important to understand that probability is always expressed as a number between 0 and 1.

• 0 means that the event cannot occur.
• 1 means that the event is certain to occur.

For example, if each question on a multiple choice exam has 3 options, and only one is correct, then the probability of guessing a correct answer is \(\frac{1}{3}\). This is because only one of the three possible outcomes is favorable. If we translate this to a percentage, it represents roughly 33.33% chance of success. Understanding probability concepts lays the groundwork for more complex topics such as binomial distributions.

Multiple Choice Questions and Guessing

Multiple choice questions (MCQs) usually have a set number of options from which only one is correct. This setup naturally leads to probability analysis, especially when the answers are guessed rather than known. For instance, consider an exam with 5 multiple choice questions, each with 3 options:

• If you know the answer to a question, you have a 100% chance of getting it right.
• But if you guess, your chance drops significantly, e.g., to \( \frac{1}{3} \).

When you're attempting to answer all questions by guessing, each correct answer is an isolated trial. This means previous or subsequent guesses don't affect the results of another. This independent nature of outcomes is key in understanding why we use binomial distribution in such scenarios. The randomness of guessing introduces a strategic probability element which is calculated through specific formulas.

The Binomial Probability Formula

The binomial probability formula is crucial in calculating the likelihood of a specific outcome in multiple choice questions when guesses are involved. It's structured to handle situations where there are a fixed number of independent trials, each with two possible outcomes: a success or a failure. This makes it perfect for exams scored through correct answers and incorrect guesses.
The formula is given by:\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]
Where:

• \(n\) is the total number of trials (questions).
• \(k\) is the number of successful trials (correct guesses).
• \(p\) is the probability of a success on an individual trial, determined by guessing correctly.
• \(1-p\) is the probability of failure.

Applying this formula helps us effectively assess probabilities for various numbers of correct answers, like determining the probability of scoring 4 or more correct out of 5 by pure guesswork. It allows a detailed breakdown of chances that can aid in understanding the outcomes of multiple choice testing scenarios.