Problem 65 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The mean and the variance of a binomial distribution are 4 and 2 respectively. Then the probability of 2 successes is [2004] (a) \(\frac{28}{256}\) (b) \(\frac{219}{256}\) (c) \(\frac{128}{256}\) (d) \(\frac{37}{256}\)

Step-by-Step Solution

Answer

The probability of 2 successes is option (a) \( \frac{28}{256} \).

Step 1: Understanding the Binomial Distribution

A binomial distribution is defined by the parameters \( n \) (number of trials) and \( p \) (probability of success in each trial). The mean \( \mu \) of a binomial distribution is given by \( \mu = np \) and the variance \( \sigma^2 \) is given by \( \sigma^2 = np(1-p) \). We're given \( \mu = 4 \) and \( \sigma^2 = 2 \).

Step 2: Establish Equations from Mean and Variance

From the mean equation, we have: \( np = 4 \). From the variance equation, we have: \( np(1-p) = 2 \). We will use these equations to find \( n \) and \( p \).

Step 3: Solve for Probability \( p \)

Substitute \( np = 4 \) into the variance equation:\[ np(1-p) = 2 \Rightarrow 4(1-p) = 2 \]Solving gives:\[ 4 - 4p = 2 \Rightarrow 4p = 2 \Rightarrow p = \frac{1}{2} \]

Step 4: Solve for Number of Trials \( n \)

Now that we have \( p = \frac{1}{2} \), substitute back into \( np = 4 \):\[ n \cdot \frac{1}{2} = 4 \Rightarrow n = 8 \]

Step 5: Use Binomial Formula for Probability of 2 Successes

The probability of exactly \( k \) successes in \( n \) trials is given by:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]For 2 successes out of 8 trials with \( p = \frac{1}{2} \):\[ P(X = 2) = \binom{8}{2} \left( \frac{1}{2} \right)^2 \left( \frac{1}{2} \right)^{8-2} = \binom{8}{2} \left( \frac{1}{2} \right)^8 \]

Step 6: Compute the Binomial Coefficient and Probability

Compute the binomial coefficient:\[ \binom{8}{2} = \frac{8 imes 7}{2 imes 1} = 28 \]Therefore:\[ P(X = 2) = 28 \times \frac{1}{256} = \frac{28}{256} \]

Step 7: Select the Correct Answer

The probability of 2 successes is \( \frac{28}{256} \). Match this with the options given, the answer is (a) \( \frac{28}{256} \).

Key Concepts

Mean

In a binomial distribution, the mean is a measure of central tendency that shows what you can expect as an average number of successes in a series of trials. It's determined by the formula \( \mu = np \), where \( n \) is the number of trials and \( p \) is the probability of success on a single trial.

For example, if you flip a coin 8 times, and the probability of getting heads on a single flip is \( \frac{1}{2} \), then you can use the formula \( \mu = 8 \times \frac{1}{2} = 4 \). This indicates that, on average, 4 heads can be expected.

Understanding the mean helps you predict outcomes when facing uncertainty, like estimating the average number of patients visiting a clinic in a week based on past data.

Variance

Variance in a binomial distribution describes the spread of the distribution or how much the outcomes deviate from the mean. It's given by the formula \( \sigma^2 = np(1-p) \).

The term \( (1-p) \) represents the probability of failure in each trial, providing insight into not only how often successes occur but also when they don't. For instance, if the mean is 4 and variance is 2, this shows both a balanced number of successes and a notable spread of results.

Calculating variance is essential in predicting unexpected deviations. For example, a high variance could indicate a wider range of customer purchases, which could influence stocking decisions in a store.

Probability of Success

The probability of success in a binomial distribution is fundamental for predicting outcomes. It is denoted by \( p \) and represents the likelihood of achieving a success in a single trial.

For instance, when we say the probability of rolling a four with a die is \( \frac{1}{6} \), this is our \( p \) value. Understanding this concept allows for calculations of more complex probabilities, like the probability of rolling a four twice in three rolls.

In the given problem, once the mean and variance were found, determining \( p \) was crucial to finding the probability of exactly 2 successes in 8 trials using the binomial formula. Mastering these calculations allows for accurate predictions of outcomes in real-world scenarios, like determining the chance of delivering a package by a certain deadline.