Problem 4 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If the probability that a student is not a swimmer is \(1 / 5\), then the probability that out of 5 students 1 is swimmer is (a) \({ }^{5} C_{1}\left(\frac{4}{5}\right)^{4}\left(\frac{1}{5}\right)\) (b) \({ }^{5} C_{1} \frac{4}{5}\left(\frac{1}{5}\right)^{4}\) (c) \(\frac{4}{5}\left(\frac{1}{5}\right)^{4}\) (d) None of these

Step-by-Step Solution

Answer

The correct answer is option (a).

Step 1: Identify Known Values

We know that the probability of a student not being a swimmer is \( \frac{1}{5} \). This implies the probability of a student being a swimmer is \( 1 - \frac{1}{5} = \frac{4}{5} \).

Step 2: Define the Binomial Probability

We need to find the probability that out of 5 students, 1 is a swimmer. This is a binomial probability problem where \( n = 5 \) trials, \( k = 1 \) success, the probability of success (swimmer) is \( p = \frac{4}{5} \), and the probability of failure (not a swimmer) is \( q = \frac{1}{5} \).

Step 3: Use Binomial Probability Formula

The binomial probability formula is given by: \[ P(X = k) = { }^n C_k p^k q^{n-k} \]Substitute the values we have:\[ P(X = 1) = { }^5 C_1 \left( \frac{4}{5} \right)^1 \left( \frac{1}{5} \right)^{5-1} \]

Step 4: Calculate the Combination Term

Calculate the combination term for choosing \( k = 1 \) from \( n = 5 \):\[ { }^5 C_1 = \frac{5!}{1!(5-1)!} = 5 \]

Step 5: Substitute and Simplify

Substitute \( { }^5 C_1 = 5 \) in the equation:\[ P(X = 1) = 5 \left( \frac{4}{5} \right) \left( \frac{1}{5} \right)^4 \]This matches with option (a):\[ { }^5 C_1 \left( \frac{4}{5} \right)^4 \left( \frac{1}{5} \right) \]

Key Concepts

Probability of Success

Understanding the probability of success is vital in solving any binomial probability problem. In this exercise, the term "success" refers to the event we are interested in, which is a student being a swimmer. The complementary event, a student not being a swimmer, is referred to as a "failure."

• Here, the probability of a student not being a swimmer is \( \frac{1}{5} \).
• To find the probability of success (i.e., a student being a swimmer), subtract the probability of failure from 1. This is: \[ 1 - \frac{1}{5} = \frac{4}{5} \]

This probability tells us that each student has a 80% chance of being a swimmer, assuming the given conditions. Understanding these probabilities forms the basis for calculating the probability of various numbers of swimmers among the group of students.

Binomial Coefficient

In binomial probability, the binomial coefficient represents the number of ways to choose a specific number of successes (or certain events) out of a total number of trials. It is symbolized as \( { }^n C_k \) and calculated using formulas from combinatorics.

• The general formula is: \[ { }^n C_k = \frac{n!}{k!(n-k)!} \]
• In our problem, we have \( n = 5 \) total students, and we are looking for \( k = 1 \) swimmer. So, the calculation is: \[ { }^5 C_1 = \frac{5!}{1!(5-1)!} = 5 \]

This result means there are 5 different ways in which 1 student can be a swimmer out of 5 students. Calculating this coefficient is crucial for forming the binomial probability formula.

Probability Calculation Steps

Calculating the probability of exactly 1 student being a swimmer out of 5 involves several steps using the binomial probability formula.
Here's a comprehensive guide to those steps:

• Identify that this is a binomial probability problem with \( n = 5 \) trials (students), \( k = 1 \) success (swimmer), \( p = \frac{4}{5} \) probability of success, and \( q = \frac{1}{5} \) probability of failure.
• The formula is: \[ P(X = k) = { }^n C_k p^k q^{n-k} \]
• Substitute the known values into the formula: \[ P(X = 1) = { }^5 C_1 \left( \frac{4}{5} \right)^1 \left( \frac{1}{5} \right)^{5-1} \]
• Calculate the combination term, which we found to be 5, and perform the power calculations.
• Finally, simplify to find: \[ P(X = 1) = 5 \left( \frac{4}{5} \right) \left( \frac{1}{5} \right)^4 \]

Following these steps, we arrive at option (a) as the correct answer. By understanding each iterative step, you can tackle more complex binomial distribution problems with confidence.