Problem 169 - A Complete Resource Book in Mathematics for JEE Main

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) \(\frac{13}{3^{5}}\) (B) \(\frac{11}{3^{5}}\) (C) \(\frac{10}{3^{5}}\) (D) \(\frac{17}{3^{5}}\)

Step-by-Step Solution

Answer

The correct answer is (B) \( \frac{11}{3^5} \).

Step 1: Understand the Problem

We need to find the probability of a student guessing and getting at least 4 out of 5 questions correct on a multiple-choice exam, where each question has 3 options. Only one option is correct for each question.

Step 2: Identify the Probabilities

For each question, the probability of choosing the correct answer by guessing is \( \frac{1}{3} \), and the probability of choosing an incorrect answer is \( \frac{2}{3} \).

Step 3: Use Binomial Probability Formula

The situation fits a binomial distribution with \( n = 5 \) (questions) and probability of success \( p = \frac{1}{3} \). The probability of getting exactly \( k \) correct answers is given by the binomial formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

Step 4: Calculate the Probability of 4 Correct Answers

For \( k = 4 \), calculate: \[ P(X = 4) = \binom{5}{4} \left( \frac{1}{3} \right)^4 \left( \frac{2}{3} \right)^{1} \] This simplifies to: \[ 5 \cdot \frac{1}{81} \cdot \frac{2}{3} = \frac{10}{243} \]

Step 5: Calculate the Probability of 5 Correct Answers

For \( k = 5 \), calculate: \[ P(X = 5) = \binom{5}{5} \left( \frac{1}{3} \right)^5 = \frac{1}{243} \]

Step 6: Add Probabilities of Getting 4 or 5 Correct Answers

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

Step 7: Compare with Options

The result from our calculation is \( \frac{11}{3^5} \). Compare this to the given options: (A) \( \frac{13}{3^5} \), (B) \( \frac{11}{3^5} \), (C) \( \frac{10}{3^5} \), (D) \( \frac{17}{3^5} \). The correct answer is option (B).

Key Concepts

Binomial Distribution

In probability theory, a binomial distribution is a common and essential concept. This distribution applies to scenarios where there are two possible outcomes: success or failure. Binomial distributions are highly applicable in real-world scenarios, such as deciding between two options, like correct or incorrect answers on a test. Binomial distributions have two parameters:

• \( n \): number of trials (in our case, the number of questions)
• \( p \): probability of success on a single trial (probability of guessing correctly)

The binomial distribution provides a framework to calculate the probability of getting a certain number of successes (correct answers) in a specific number of trials. Understanding this concept allows us to compute the likelihood of different outcomes using the binomial probability formula.

Multiple Choice Examination

A multiple choice exam can be an intriguing application of probability theory. It typically involves questions with several answer options, of which only one is correct. In this exercise, we are dealing with a test of five questions, each offering three answer choices.This scenario highlights a few crucial aspects:

• Each question is independent of the others. Answering one question correctly doesn't affect the others.
• The probability of correctly guessing an answer amongst three choices is \( \frac{1}{3} \), since there is only one right answer.
• The examination is structured so that random guessing can lead to a certain distribution of correct and incorrect answers.

This independence and equal probability framework makes multiple choice examinations perfectly suited to binomial distribution analysis.

Binomial Probability Formula

The binomial probability formula is an essential tool for calculating the likelihood of a specific event occurring a certain number of times. It is given by:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Where:

• \( n \) is the number of trials (questions in the exam).
• \( k \) is the number of successes we want to find the probability for (correct answers).
• \( p \) is the probability of a success on any given trial.
• \( \binom{n}{k} \) represents the number of combinations of \( n \) trials taken \( k \) at a time.

This formula helps calculate the probability of all possible scenarios in a binomial setting, like getting 4 or 5 correct answers by guessing in a 5-question exam.

Correct and Incorrect Answers Probability

Probability calculations for correct and incorrect answers are crucial in understanding how often a student might succeed by guessing. In the context of a multiple choice exam:- The probability of correctly answering by guessing is \( \frac{1}{3} \).- Conversely, the probability of an incorrect answer in a single question is \( \frac{2}{3} \).Using these probabilities, we can employ the binomial probability formula to assess the student's likelihood of achieving a specific number of correct answers. For example, in this exercise, one could calculate the probability of getting exactly 4 or 5 correct answers, and by summing these, determine the probability of getting at least 4 right answers.Understanding these probabilities empowers students to analyze their chances of success and can also inform strategies for test-taking and study.