Solution & Explanation

Understanding Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success. This distribution is particularly useful in scenarios where there are only two possible outcomes, often labeled as "success" and "failure." The binomial distribution is characterized by two parameters:

• n: The number of trials.
• p: The probability of success on each trial.

Expected Value of a Binomial Distribution

The expected value (or mean) of a binomial distribution is a measure of the central tendency of the distribution. It gives us a sense of the average number of successes we can expect over many repetitions of the experiment.

Formula

The expected value of a binomial distribution is calculated using the formula:

$E(X) = n \times p$

Where:

• E(X) represents the expected value of the random variable X.
• n is the number of trials.
• p is the probability of success in each trial.

Explanation

The formula for the expected value is derived from the properties of the binomial distribution. Since each trial is independent, the expected number of successes in each trial is simply the probability of success, p. Therefore, the expected number of successes over n trials is n times p.

Example

Consider a scenario where you are flipping a fair coin five times, and you want to find the expected number of heads (successes):

• n = 5 (number of trials)
• p = 0.5 (probability of getting heads in each trial)

Using the formula:

$E(X) = 5 \times 0.5 = 2.5$

This means that, on average, you can expect to get 2.5 heads when flipping the coin five times.

Why It Matters

Understanding the expected value of a binomial distribution is crucial for predicting outcomes in various fields such as finance, medicine, and quality control. It helps in making informed decisions by providing a statistical measure of the average outcome over many trials.

Key Points

• The binomial distribution is applicable when there are two possible outcomes per trial (success/failure).
• The expected value provides a measure of the average number of successes.
• The formula $E(X) = n \times p$ is simple but powerful in predicting outcomes over repeated trials.

In summary, the expected value of a binomial distribution is a foundational concept in probability and statistics, offering insights into the average behavior of a random process characterized by binary outcomes.