Binomial Distribution

Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in $n$ independent Bernoulli trials, where the probability of success in each trial is $p$.

It is used when:

1. Each trial has only two possible outcomes (success or failure).
2. The probability of success ($p$) remains constant for all trials.
3. The trials are independent of each other.

Formula

The probability of observing exactly $k$ successes is given by the formula:

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

Where:

• $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.
• $n$ is the total number of trials.
• $k$ is the number of successes.
• $p$ is the probability of success.
• $(1-p)$ is the probability of failure.

Mean and Variance

The mean and variance of a binomial distribution are:

$$\text{Mean: } \mu = n \times p$$ $$\text{Variance: } \sigma^2 = n \times p \times (1-p)$$

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Steps to Solve Binomial Distribution Problems

1. Define the Variables: Identify $n$, $p$, and $k$
2. Apply the Formula: Use the binomial probability formula to calculate $P(X = k)$
3. Simplify the Expression: Compute factorials and powers as needed.
4. Interpret the Result: Relate the calculated probability to the context of the problem.

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Worked Examples

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Summary

• The binomial distribution is used for problems involving $n$ independent trials with two outcomes (success or failure).
• The probability formula is:

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

• Mean: $\mu = n \times p$
• Variance: $\sigma^2 = n \times p \times (1-p)$
• Common applications include quality control, surveys, and games of chance.