Binomial Distribution: Explanation

The binomial distribution is a probability distribution that summarizes the likelihood of a given number of successes in a fixed number of independent experiments, each with the same probability of success.

Key characteristics:

1. Fixed number of trials (n): The experiment is repeated a set number of times.
2. Only two outcomes per trial: Success (e.g., heads in a coin toss) or failure (e.g., tails).
3. Constant probability (p): The probability of success remains the same in each trial.
4. Independence: The outcome of one trial does not affect the outcome of another.

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:

• $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the number of ways to choose $k$ successes from $n$ trials.
• $p$ is the probability of success.
• $1-p$ is the probability of failure.
• $k$ is the number of successes (0 to $n$).

---

Example of Binomial Distribution

Scenario: A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads?

Parameters:

• $n = 5$ (number of trials)
• $k = 3$ (number of successes)
• $p = 0.5$ (probability of heads)

Using the formula:

$P(X = 3) = \binom{5}{3} (0.5)^3 (1-0.5)^{5-3}$

1. Calculate $\binom{5}{3}$: $\binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \cdot 4}{2 \cdot 1} = 10$

2. Calculate probabilities: $(0.5)^3 = 0.125, \quad (1-0.5)^2 = (0.5)^2 = 0.25$

3. Combine terms: $P(X = 3) = 10 \cdot 0.125 \cdot 0.25 = 0.3125$

Result: The probability of getting exactly 3 heads in 5 tosses is 0.3125 (31.25%).

---

Why Use Binomial Distribution?

It’s useful in real-life scenarios where you’re interested in counting the number of successes, such as:

• Quality control: Number of defective items in a batch.
• Surveys: Number of people favoring a policy.
• Medicine: Patients responding positively to a drug.

---

Follow-Up Questions:

1. How can the binomial distribution be applied in quality control processes?
2. What are some limitations of the binomial distribution?
3. How does the shape of a binomial distribution change as $p$ varies?
4. What is the difference between binomial and Poisson distributions?
5. Can you explain how to calculate cumulative probabilities for binomial distribution?

---

Tip:

If $n$ is large and $p$ is not extremely close to 0 or 1, the binomial distribution can be approximated by a normal distribution for easier calculations.