**Definition**

The **binomial distribution** is defined as the discrete probability distribution of obtaining exactly n successes out of N trials. Here, a success is denoted by the outcome we are looking to find the probability for, and it occurs with a probability p. The probability of failure is denoted by q, where q = 1 - p.

In mathematical terms, the probability P_{p}(n|N) of obtaining exactly n successes in N trials is given by:

P_{p}(n|N) = (N choose n) * p^{n} * q^{(N - n)}

Where:

- (N choose n) is the binomial coefficient, representing the number of ways to choose n successes from N trials.
- p is the probability of success on any given trial.
- q is the probability of failure on any given trial.

For a deeper understanding of probability, refer to the basics of probability.

**Mean**

The mean, or expected value, of a binomial distribution provides the average number of successes we can expect in N trials. It's given by:

**Mean = N * p**

For instance, if you were to flip a fair coin (where p = 0.5) 10 times (N = 10), the expected number of heads would be:

Mean = 10 * 0.5 = 5

This implies that, on average, you'd expect to get 5 heads when flipping the coin 10 times.

Understanding the normal distribution can also help in grasping different types of probability distributions.

**Variance**

Variance measures the spread or dispersion of a set of data points. For the binomial distribution, the variance is:

**Variance = N * p * q**

Using our coin flipping example, the variance would be:

Variance = 10 * 0.5 * 0.5 = 2.5

This gives an idea of the spread of the possible number of heads we might get when flipping the coin 10 times.

The concept of correlation coefficient is also important when discussing data spread and relationships.

**Properties**

The binomial distribution has several key properties:

1. **Discreteness**: The binomial distribution is discrete, meaning it can only take on a finite number of values. In the context of our coin flipping example, you can't get 2.5 heads; you can only get a whole number of heads.

2. **Fixed Number of Trials**: The number of trials N is fixed in advance. This means you decide beforehand how many times you'll flip the coin, take a quiz, etc.

3.** Independent Trials**: Each trial is independent of the others. The outcome of one coin flip doesn't affect the outcome of the next coin flip.

4.** Constant Probability**: The probability of success p remains constant from trial to trial. Every time you flip a fair coin, there's always a 0.5 chance of getting heads.

5. **Two Possible Outcomes**: Each trial has only two possible outcomes, often termed success and failure.

For more on how data can be visualised, check out the line of best fit.

**Example Question**

Imagine you're taking a multiple-choice quiz with 20 questions. Each question has 4 options, and you're guessing on all of them. What's the expected number of questions you'd get right, and what's the variance?

**Solution**:

Here, N = 20 (since there are 20 questions) and p = 0.25 (since there's a 1 in 4 chance of guessing correctly).

Mean = N * p Mean = 20 * 0.25 = 5

Variance = N * p * q Variance = 20 * 0.25 * 0.75 = 3.75

So, if you were to guess on all the questions, you'd expect to get 5 right on average, with a variance of 3.75.

To further explore the connection between probability and outcomes, consider studying Bayes' Theorem.

## FAQ

The binomial distribution is crucial in real-world applications because many phenomena can be modelled as a series of independent trials with two possible outcomes. For example, quality control in manufacturing often involves checking if items are defective or not, medical trials might look at the presence or absence of side effects, and marketing campaigns might evaluate the success or failure of customer responses. By understanding and applying the binomial distribution, businesses and researchers can make informed decisions, predict outcomes, and assess risks. It provides a mathematical framework to analyse and interpret results in various fields.

No, the binomial distribution specifically models scenarios with two possible outcomes for each trial, often termed as success and failure. If you have a situation with more than two outcomes, you would need to look at other distributions. For instance, the multinomial distribution is an extension of the binomial distribution that deals with experiments with more than two possible outcomes. It's essential to choose the appropriate distribution based on the nature of the experiment and the number of possible outcomes to get accurate results.

The value of 'p', the probability of success, plays a significant role in shaping the binomial distribution. When 'p' is 0.5, the distribution is symmetric, resembling a bell shape. This means that the probabilities of successes and failures are equal. However, as 'p' deviates from 0.5, the distribution becomes skewed. If 'p' is greater than 0.5, the distribution skews to the right, indicating a higher probability of successes. Conversely, if 'p' is less than 0.5, the distribution skews to the left, showing a higher likelihood of failures. Understanding the shape and skewness helps in visualising and interpreting the distribution in various contexts.

The binomial distribution and the normal distribution are both probability distributions, but they have distinct characteristics. The binomial distribution is discrete and models the number of successes in a fixed number of independent trials with a constant probability of success. It is defined by two parameters: the number of trials (n) and the probability of success (p). On the other hand, the normal distribution is continuous and is defined by its mean (µ) and standard deviation (σ). It describes data that clusters around a mean or central value. While the binomial distribution has a finite range, the normal distribution extends infinitely in both directions. However, for large sample sizes and certain conditions, the binomial distribution can be approximated by the normal distribution.

The term "binomial" originates from the Latin words "bi-" meaning two and "nomial" meaning term. In the context of the binomial distribution, it refers to the two possible outcomes in each trial: success and failure. The binomial distribution models the probability of achieving a specific number of successes in a fixed number of trials, each with the same probability of success. The mathematical representation of the binomial distribution involves binomial coefficients, which further ties into the name. These coefficients represent the number of ways to choose a set number of successes from a given number of trials.

## Practice Questions

The situation described follows a binomial distribution. Given that n = 200 (number of trials) and p = 0.03 (probability of success, which in this context is a bulb being defective):

Expected number of defective light bulbs (mean) = n * p = 200 * 0.03 = 6

Variance = n * p * (1-p) = 200 * 0.03 * 0.97 = 5.82

Thus, in a sample of 200 light bulbs, we'd expect 6 to be defective with a variance of 5.82.

This is a binomial distribution problem where n = 50 (number of trials) and p = 0.25 (probability of success, which in this case is guessing the answer correctly).

The probability P of getting exactly 10 questions correct is given by:

P = (n choose k) * p^{k} * (1-p)^{(n-k)}

Where k is the number of successes, which is 10 in this case.

P = (50 choose 10) * 0.25^{10} * 0.75^{40}

Using the binomial coefficient formula and simplifying, we get:

P approximately equals 0.098

So, the probability that the student gets exactly 10 questions correct by guessing is approximately 0.098 or 9.8%.