Lesson 3.2: The Binomial Distribution

Introduction

In this lesson, we will explore the binomial distribution, a fundamental concept in statistics that models the number of successes in a fixed number of independent Bernoulli trials. Understanding when to use the binomial distribution, how to evaluate probabilities, and interpreting its parameters are essential skills in A-Level Statistics. By the end of this lesson, you will be able to:

• Recognise when a binomial model is appropriate, including modelling assumptions in real situations.
• Evaluate or read binomial probabilities using a calculator, formula, or tables.
• Calculate and interpret the mean and variance of a binomial distribution.
• State the conditions for a binomial model and judge whether they hold in a given context.
• Find binomial probabilities, including cumulative probabilities, using a calculator.

What is a Binomial Distribution?

The binomial distribution describes the number of successes in a fixed number of trials, each with the same probability of success. It's characterized by two parameters:

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

The random variable $ X $ representing the number of successes in $ n $ trials is said to follow a binomial distribution, denoted by:

$$\ X \sim B(n, p)$$

Conditions for a Binomial Model

To correctly apply the binomial distribution, the following conditions must be met:

1. Fixed number of trials: The number of trials $ n $ is predetermined and does not change.
2. Two possible outcomes: Each trial results in a success or failure.
3. Constant probability: The probability of success $ p $ remains the same for each trial.
4. Independent trials: The outcome of one trial does not affect the outcome of another.

Examples of Binomial Situations

• Coin Tossing: Flipping a coin 10 times and counting the number of heads.
• Manufacturing Quality Control: Inspecting a batch of 20 items and counting how many are defective (assuming a constant defect rate).

Evaluating Binomial Probabilities

The probability of obtaining exactly $ k $ successes in $ n $ trials is given by the formula:

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

where $ \binom{n}{k} $ is the binomial coefficient, calculated as:

$$\binom{n}{k} = \frac{n!}{k!(n - k)!}$$

Worked Example 1: Tossing a Coin

Suppose you toss a fair coin 8 times. What is the probability of getting exactly 3 heads?

Here, we have:

• $ n = 8 $ (the number of tosses)
• $ p = 0.5 $ (the probability of getting heads)
• $ k = 3 $ (the number of successes we are interested in)

First, we calculate the binomial coefficient:

$$\binom{8}{3} = \frac{8!}{3!(8 - 3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$$

Now, plug this into the probability formula:

$$P(X = 3) = 56 \times (0.5)^3 \times (0.5)^{8 - 3} = 56 \times (0.5)^8 = 56 \times \frac{1}{256} = \frac{56}{256} = 0.21875$$

So the probability of getting exactly 3 heads in 8 coin tosses is $ 0.21875 $.

Cumulative Probabilities

In many scenarios, you may be interested in finding the probability of getting up to $ k $ successes, called cumulative probability. This can be calculated using:

$$P(X \leq k) = \sum_{i=0}^{k} P(X = i)$$

Alternatively, most calculators and statistical software can compute cumulative probabilities directly.

Worked Example 2: Cumulative Probability

Continuing from our earlier example with 8 coin tosses, what is the probability of getting at most 3 heads?

Using our previous calculations, we find:

$$P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)$$

You can calculate each term using the binomial formula, or find them directly using your calculator. After thorough calculations:

• $ P(X = 0) \approx 0.0391 $
• $ P(X = 1) \approx 0.1563 $
• $ P(X = 2) \approx 0.2188 $
• $ P(X = 3) \approx 0.21875 $

Adding these up gives:

$$P(X \leq 3) \approx 0.0391 + 0.1563 + 0.21875 + 0.21875 \approx 0.6329$$

Thus, the probability of getting at most 3 heads is approximately $ 0.6329 $.

Mean and Variance of the Binomial Distribution

The mean $ \mu $ and variance $ \sigma^2 $ of a binomial distribution are calculated as follows:

• Mean: $$\mu = n \cdot p$$
• Variance: $$\sigma^2 = n \cdot p \cdot (1 - p)$$

Worked Example 3: Mean and Variance Calculation

Using our previous example of 8 coin tosses where $ p = 0.5 $:

• Mean: $$\mu = 8 \cdot 0.5 = 4$$
• Variance: $$\sigma^2 = 8 \cdot 0.5 \cdot (1 - 0.5) = 8 \cdot 0.5 \cdot 0.5 = 2$$

Thus, for our 8 coin tosses, the mean number of heads is 4, and the variance is 2.

Conclusion

In this lesson, we unpacked the binomial distribution, understanding its definitions, applications, and calculations. We learned to:

• Recognise conditions for using a binomial distribution model.
• Calculate probabilities using the binomial formula and cumulative distributions.
• Determine the mean and variance of a binomial distribution.

These concepts are foundational for further studies in statistics and will aid in interpreting data in various real-life scenarios.

Study Notes

• The binomial distribution models successes in a fixed number of trials with constant probability.
• Conditions for a binomial model: fixed trials, two outcomes, constant probability, and independence of trials.
• Probability of exactly $ k $ successes: $$P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$$
• Cumulative probability $ P(X \leq k) $ involves summing probabilities up to $ k $.
• Mean of binomial: $\mu = n \cdot p$; Variance of binomial: $\sigma^2 = n \cdot p \cdot (1 - p)$.