Problem 169 - A Complete Resource Book in Mathematics for JEE Main 2017

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 probability is \( \frac{11}{3^5} \) which is option (B).

Step 1: Understand the problem

The problem involves a multiple choice exam with 5 questions and 3 answer choices each. We need to calculate the probability of getting 4 or more correct answers by guessing. This means we need the probabilities of getting exactly 4 correct answers and all 5 correct answers.

Step 2: Define the probability of correct answer

Since each question has 3 choices and only one is correct, the probability of guessing a question correctly is \( \frac{1}{3} \). Similarly, the probability of guessing incorrectly is \( \frac{2}{3} \).

Step 3: Calculate the probability for exactly 4 correct answers

To calculate the probability of getting exactly 4 correct answers out of 5, we use the binomial probability formula: \[P(X = 4) = \binom{5}{4} \left(\frac{1}{3}\right)^4 \left(\frac{2}{3}\right)^{1}.\]Calculating this gives:\[P(X = 4) = 5 \times \frac{1}{81} \times \frac{2}{3} = \frac{10}{243}.\]

Step 4: Calculate the probability for exactly 5 correct answers

For exactly 5 correct answers, the formula is: \[P(X = 5) = \binom{5}{5} \left(\frac{1}{3}\right)^5 \left(\frac{2}{3}\right)^{0}.\]Calculating this gives:\[P(X = 5) = 1 \times \frac{1}{243} = \frac{1}{243}.\]

Step 5: Calculate total probability for 4 or more correct answers

Add the probabilities from Step 3 and Step 4:\[P(X \geq 4) = P(X = 4) + P(X = 5) = \frac{10}{243} + \frac{1}{243} = \frac{11}{243}.\]

Step 6: Compare with answer choices

The calculated probability \( \frac{11}{243} \) matches answer choice (B), which confirms our solution.

Key Concepts

Binomial Probability

Binomial probability is a fundamental concept in probability theory. It is used to determine the likelihood of a specific number of successes in a sequence of independent trials, each with two possible outcomes. This is where it gets its name, 'bi' meaning two. In this context, the outcomes are usually referred to as "success" and "failure."

Here’s a simple breakdown of the binomial probability formula:

• First, identify the number of trials, denoted as \( n \).
• Determine the probability of success on a single trial, \( p \).
• The formula for binomial probability is \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( k \) is the number of successes.
• \( \binom{n}{k} \), known as a binomial coefficient, looks like this: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).

In practical applications, such as multiple choice questions, each question represents a trial, and each choice a possible outcome. Using the binomial probability formula simplifies calculations when determining the probability of guessing a certain number of correct answers.

Multiple Choice Questions

Multiple choice questions (MCQs) are a common testing format used in many exams. Typically, each question offers several possible answers, with only one of them being correct. Let's delve into how probability interacts with this form of questioning.

In an exam setting featuring multiple choice questions:

• Each question is an independent event.
• If the student is guessing, they have an equal chance of selecting any of the choices.
• With three options and one correct answer, the probability of guessing correctly is \( \frac{1}{3} \).
Understanding the nature of MCQs is essential for calculating probabilities related to guessing, especially when employing binomial probability. Ensuring you know the number of potential answers and identifying which is correct can transform guessing from a hit-and-miss ordeal into a calculated probability challenge. Moreover, knowing how to compute using binomial probability helps us to evaluate the chance of guessing multiple questions correctly, which can be especially useful in a gaming context where probability matters.

Probability of Guessing Correctly

In probability theory, understanding the mechanics of guessing is crucial in scenarios where people resort to making educated guesses, such as in a multiple choice examination.

To calculate the probability of guessing correctly on one question:

• Identify the total number of choices (let's say, 3 for each question).
• The chance of randomly selecting the correct answer is \( \frac{1}{3} \), and conversely, the probability of selecting incorrectly is \( \frac{2}{3} \).

For multiple questions, such as 5 questions in this instance:

• We use the binomial probability framework to calculate probabilities for different numbers of correct guesses.
• Adding probabilities helps us find the likelihood of more than one possibility, e.g., guessing at least 4 questions correctly.
• In our example, this involves computing both \( P(X = 4) \) and \( P(X = 5) \), and summing these values.

Mastering these computations empowers students in understanding their odds better. For instance, when guessing all answers on a 5-question MCQ test, knowing that the theoretical probability of getting 4 or more correct just by chance is \( \frac{11}{243} \) can guide their testing strategy, especially when under time pressure or in topics they find challenging.