Finding Binomial Distribution Parameters from Mean and Variance

A binomial distribution is defined by two parameters: \(n\), the number of trials, and \(p\), the probability of success on a single trial. We are given the mean and the variance of a binomial distribution and need to find the probability \(P(X=0)\).

The formulas for the mean (\(\mu\)) and variance (\(\sigma^2\)) of a binomial distribution are:

• Mean: \(\mu = np\)
• Variance: \(\sigma^2 = np(1-p)\)

We are given that the mean is 2 and the variance is 1. So, we have the following system of equations:

1. \(np = 2\)
2. \(np(1-p) = 1\)

We can use these equations to find the values of \(n\) and \(p\).

Calculating Binomial Distribution Parameters n and p

Substitute the expression for the mean (\(np\)) from equation (1) into equation (2):

\(2(1-p) = 1\)

Now, solve for \(p\):

\(1-p = \frac{1}{2}\)

\(p = 1 - \frac{1}{2}\)

\(p = \frac{1}{2}\)

Now that we have the value of \(p\), substitute it back into equation (1) to find \(n\):

\(n \times \left(\frac{1}{2}\right) = 2\)

\(n = 2 \times 2\)

\(n = 4\)

So, the parameters of this binomial distribution are \(n=4\) and \(p=\frac{1}{2}\).

Calculating the Probability P(X = 0)

The probability mass function (PMF) for a binomial distribution \(B(n, p)\) is given by:

\(P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\)

We need to find \(P(X=0)\) for \(n=4\) and \(p=\frac{1}{2}\). Here, \(k=0\).

Substitute the values into the PMF formula:

\(P(X=0) = \binom{4}{0} \left(\frac{1}{2}\right)^0 \left(1-\frac{1}{2}\right)^{4-0}\)

\(P(X=0) = \binom{4}{0} \left(\frac{1}{2}\right)^0 \left(\frac{1}{2}\right)^4\)

Let's evaluate the components:

• \(\binom{4}{0} = \frac{4!}{0!(4-0)!} = \frac{4!}{0!4!} = \frac{4!}{1 \times 4!} = 1\) (since \(0! = 1\))
• \(\left(\frac{1}{2}\right)^0 = 1\) (any non-zero number raised to the power of 0 is 1)
• \(\left(\frac{1}{2}\right)^4 = \frac{1^4}{2^4} = \frac{1}{16}\)

Now, multiply these values together:

\(P(X=0) = 1 \times 1 \times \frac{1}{16}\)

\(P(X=0) = \frac{1}{16}\)

Thus, the probability \(P(X=0)\) for this binomial distribution is \(\frac{1}{16}\).

Revision Table: Binomial Distribution Formulas

Additional Information on Binomial Probability

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success or failure) and the probability of success is constant for every trial. These are often called Bernoulli trials.

• Trials (n): The total count of identical and independent experiments.
• Success Probability (p): The likelihood of the desired outcome in a single trial.
• Failure Probability (1-p): The likelihood of the undesired outcome in a single trial.
• Random Variable (X): Represents the number of successes in \(n\) trials. \(X\) can take integer values from 0 to \(n\).

Understanding the relationship between the mean, variance, and the parameters \(n\) and \(p\) is crucial for solving problems involving the binomial distribution when these summary statistics are provided.