Problem 34 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

In binomial probability distribution, mean is 3 and standard deviation is \(3 / 2\). Then the probability distribution is (a) \(\left(\frac{3}{4}+\frac{1}{4}\right)^{12}\) (b) \(\left(\frac{1}{4}+\frac{3}{4}\right)^{12}\) (c) \(\left(\frac{1}{4}+\frac{3}{4}\right)^{9}\) (d) \(\left(\frac{3}{4}+\frac{1}{4}\right)^{9}\)

Step-by-Step Solution

Answer

The correct option is (b) \( \left(\frac{1}{4} + \frac{3}{4}\right)^{12} \).

Step 1: Understand the Problem

We are given a binomial distribution with a mean of 3 and a standard deviation of \( \frac{3}{2} \). We need to determine which probability distribution matches these parameters from the given options.

Step 2: Recall the Formula for Mean

The mean \( \mu \) of a binomial distribution is given by \( \mu = n \times p \), where \( n \) is the number of trials and \( p \) is the probability of success in a single trial.

Step 3: Recall the Formula for Standard Deviation

The standard deviation \( \sigma \) of a binomial distribution is given by \( \sigma = \sqrt{n \times p \times (1-p)} \). We have that \( \sigma = \frac{3}{2} \).

Step 4: Set Up the Equations

Using \( \mu = 3 \), we have \( n \times p = 3 \). Using \( \sigma = \frac{3}{2} \), we have \( \sqrt{n \times p \times (1-p)} = \frac{3}{2} \).

Step 5: Interpret the Mean Equation

From \( n \times p = 3 \), express \( p \) as \( p = \frac{3}{n} \).

Step 6: Substitute into the Standard Deviation Equation

Substitute \( p = \frac{3}{n} \) into the equation for standard deviation: \[\sqrt{n \times \frac{3}{n} \times \left(1-\frac{3}{n}\right)} = \frac{3}{2}\]

Step 7: Simplify the Equation

Simplify to: \[\frac{3}{2} = \sqrt{3 - \frac{9}{n}}\] Square both sides to get: \[\left(\frac{3}{2}\right)^2 = 3 - \frac{9}{n}\] which simplifies to: \[\frac{9}{4} = 3 - \frac{9}{n}\]

Step 8: Solve for n

Rearrange the equation: \[\frac{9}{n} = 3 - \frac{9}{4}\] Simplifying gives: \( \frac{9}{n} = \frac{3}{4} \). Solve for \( n \) to get \( n = 12 \).

Step 9: Find p

Now substitute \( n = 12 \) back into \( p = \frac{3}{n} \) to get \( p = \frac{3}{12} = \frac{1}{4} \).

Step 10: Determine the Probability Distribution

Thus, \( n = 12 \) and \( p = \frac{1}{4} \). The other probability \( q = 1-p = \frac{3}{4} \). The distribution is \( (\frac{1}{4} + \frac{3}{4})^{12} \).

Step 11: Choose the Correct Option

Compare the expression to the given options. Option (b) matches: \( \left(\frac{1}{4} + \frac{3}{4}\right)^{12} \).

Key Concepts

Mean of Binomial Distribution

The mean of a binomial distribution is a measure of its central tendency, representing the expected outcome over numerous trials. It is calculated using the formula:\[\mu = n \times p\]where \(n\) is the number of trials, and \(p\) is the probability of success in each trial. In our problem, the mean is given as 3. This means, on average, we can expect 3 successes out of the total number of trials, if the experiment is repeated many times. The mean helps us understand what outcome to typically expect when conducting a series of binomial experiments.

Standard Deviation of Binomial Distribution

The standard deviation in a binomial distribution quantifies the variation or spread of outcomes around the mean. It is computed as:\[\sigma = \sqrt{n \times p \times (1-p)}\]For this problem, the standard deviation is given as \(\frac{3}{2}\). This tells us how much the number of successful outcomes can typically vary from the mean, 3, for a large number of trials. While the mean gives us an expected value, the standard deviation provides insights into the reliability of that expectation, giving us an idea of the range in which actual results are likely to fall.

Probability of Success in Binomial Distribution

The probability of success \(p\) in a binomial distribution indicates the likelihood of a single trial resulting in success. In the given problem, through calculations, we've found that \(p = \frac{1}{4}\). This means there is a 25% chance of success in each trial of the experiment. Understanding \(p\) is crucial because it influences the shape and characteristics of the binomial distribution, affecting both the mean and the variance. A lower or higher \(p\) can completely change the expected results and variability of outcomes.

Number of Trials in Binomial Distribution

The number of trials \(n\) in a binomial distribution represents how many times an experiment is conducted. In this exercise, we derived \(n = 12\). Having the number of trials fixed allows us to quantify the distribution's behavior through formulas for mean and standard deviation. It is an integral parameter that influences the probability distribution, dictating not only how often a successful outcome is expected (through mean) but also the spread of possible outcomes (through standard deviation). Knowing \(n\) helps in accurately defining how the binomial process unfolds over repeated experiments.