M1 Study Notes: The Binomial Distribution

Hello! Welcome to your study notes for one of the most useful topics in statistics: The Binomial Distribution. Don't worry if the name sounds complicated. It's actually a very logical tool that helps us understand situations we see every day.

Ever wondered about the chances of guessing correctly on a multiple-choice quiz? Or the probability of a basketball player making a certain number of free throws? The binomial distribution is the key to answering these kinds of questions! In this chapter, you'll learn:

  • What a "Bernoulli trial" is (the simple building block).
  • How to identify a situation that follows a binomial distribution.
  • How to use a formula to calculate probabilities.
  • How to find the mean (expected value) and variance quickly.

Let's dive in, and you'll see it's more straightforward than you think!

1. The Building Block: The Bernoulli Trial

Before we can run, we need to walk. The "walk" of the binomial distribution is something called a Bernoulli trial. It's a fancy name for a very simple idea.

A Bernoulli trial is a single experiment that has only two possible outcomes.

We usually label these outcomes as:

  • Success: The outcome we are interested in counting.
  • Failure: The other possible outcome.

Important: "Success" doesn't have to mean something good! If you're studying the probability of a product being defective, finding a "defective" product would be the "success" for that experiment. It's just a label for what you're counting.

Examples of Bernoulli Trials:
  • Flipping a coin once: Success = getting a Head, Failure = getting a Tail.
  • Taking one free throw shot: Success = making the shot, Failure = missing the shot.
  • A single item from a production line: Success = the item is faulty, Failure = the item is not faulty.

In any Bernoulli trial, we use p to represent the probability of success, and 1-p to represent the probability of failure.

Key Takeaway

A Bernoulli trial is a one-time experiment with only two outcomes: success or failure. It's the fundamental piece of the binomial distribution.

2. What is the Binomial Distribution?

So, what happens when we do a bunch of Bernoulli trials in a row? That's when we get a binomial distribution!

A binomial distribution describes the probability of getting a specific number of successes in a fixed number of independent Bernoulli trials.

Let's go back to the coin flip.
One flip is a Bernoulli trial.
Five flips in a row is a binomial experiment. We can then ask questions like, "What is the probability of getting exactly 3 heads in these 5 flips?"

The Notation: X ~ B(n, p)

This is the official way to write that a random variable follows a binomial distribution. It looks like code, but it's simple once you break it down:

  • X: This is our discrete random variable, which represents the number of successes we get. For example, X could be "the number of heads in 5 flips".
  • ~: This little squiggly line just means "is distributed as".
  • B: This stands for Binomial.
  • (n, p): These are the two "parameters" of the distribution.
    • n = the total number of trials.
    • p = the probability of success on any single trial.

Example: If we flip a fair coin 10 times and we're counting the number of heads, the notation would be X ~ B(10, 0.5). Here, n=10 and p=0.5.

3. Is it a Binomial Problem? Use the B.I.N.S. Checklist!

You can't use the binomial distribution for every problem! A situation must meet four specific conditions. A great way to remember them is the mnemonic B.I.N.S.

The B.I.N.S. Checklist:

B - Binary: Are there only two possible outcomes for each trial (Success/Failure)?
Example: Getting a '4' on a dice roll (Success) vs. not getting a '4' (Failure).

I - Independent: Is the outcome of one trial completely unaffected by the outcome of previous trials?
Example: Flipping a coin is independent. The coin doesn't remember the last result.
Non-Example: Drawing two cards from a deck *without replacement*. The probability of the second card depends on what the first card was.

N - Number of Trials: Is the number of trials, n, fixed in advance?
Example: "We will flip the coin exactly 20 times." (Fixed)
Non-Example: "We will flip the coin until we get 3 heads." (Not fixed)

S - Same Probability: Is the probability of success, p, the same for every single trial?
Example: The probability of rolling a '6' on a fair die is always 1/6, no matter how many times you roll it.

Common Mistakes to Avoid

The most common mistake is assuming a problem is binomial without checking the B.I.N.S. conditions. Always quickly go through the checklist in your head before you start calculations!

Key Takeaway

If a situation fits all four B.I.N.S. conditions, it's a binomial problem. If even one condition fails, you cannot use the binomial formula.

4. The Formula: Your Tool for Calculation

Okay, we've identified a binomial problem. Now, how do we find the probability? We use the Binomial Probability Formula.

If X ~ B(n, p), the probability of getting exactly k successes is:

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

Don't be intimidated! Let's break it down into its three simple parts:

  1. $$C_k^n$$ This part tells us the number of ways to arrange 'k' successes among 'n' trials. You've seen this in Core Maths! It's the combinations formula, which you can find on your calculator as the nCr button. (e.g., C_2^5 means "how many ways can we choose 2 items from a set of 5?")

  2. $$p^k$$ This is the probability of getting 'k' successes. If the probability of one success is 'p', then the probability of 'k' of them happening is p multiplied by itself k times.

  3. $$(1-p)^{n-k}$$ This is the probability of getting the remaining 'n-k' trials as failures. If the probability of success is 'p', then the probability of failure is '1-p'. We need 'n-k' of them.
Step-by-Step Example

Question: A basketball player has a 70% chance of making any free throw. If she takes 5 shots, what is the probability that she makes exactly 4 of them?

Step 1: Identify n, p, and k.

  • The situation is binomial (check B.I.N.S.!).
  • Number of trials, n = 5.
  • Probability of success (making a shot), p = 0.7.
  • Number of successes we want, k = 4.
So, we want to find P(X=4).

Step 2: Plug into the formula. $$P(X=4) = C_4^5 (0.7)^4 (1-0.7)^{5-4}$$

Step 3: Calculate each part.

  • $C_4^5 = 5$ (Use your calculator's nCr function)
  • $(0.7)^4 = 0.2401$
  • $(1-0.7)^{5-4} = (0.3)^1 = 0.3$

Step 4: Multiply them together. $$P(X=4) = 5 \times 0.2401 \times 0.3 = 0.36015$$ So, there is about a 36% chance she makes exactly 4 out of 5 shots.

5. Beyond 'Exactly': 'At Least' and 'At Most'

Exam questions often ask for more than just an exact number. They might ask for the probability of "at least 2 successes" or "at most 3 successes".

"At most k" successes: P(X ≤ k)
This means k successes, or k-1, or k-2, ..., all the way down to 0. You need to calculate each probability and add them up. $$P(X \le k) = P(X=0) + P(X=1) + ... + P(X=k)$$

"At least k" successes: P(X ≥ k)
This means k successes, or k+1, ..., all the way up to n. You *could* add them all up, but there's a much faster way: the complement rule.

The total probability of all outcomes is 1. So, the probability of "at least k" is the same as 1 minus the probability of "less than k". $$P(X \ge k) = 1 - P(X < k)$$ $$P(X \ge k) = 1 - [P(X=0) + P(X=1) + ... + P(X=k-1)]$$

Example: What is the probability of the basketball player making at least 4 shots (out of 5)?
"At least 4" means P(X=4) or P(X=5). $$P(X \ge 4) = P(X=4) + P(X=5)$$
We already found P(X=4) = 0.36015.
Now we find P(X=5): $$P(X=5) = C_5^5 (0.7)^5 (0.3)^0 = 1 \times 0.16807 \times 1 = 0.16807$$
So, P(X ≥ 4) = 0.36015 + 0.16807 = 0.52822.

6. Quick Facts: Mean and Variance

Sometimes, we don't need to calculate a specific probability. We just want to know the average outcome or how spread out the results are likely to be. Thankfully, the formulas for the mean and variance of a binomial distribution are very simple!

Mean or Expected Value, E(X)
This is the average number of successes you would expect to get if you ran the experiment many, many times. $$E(X) = np$$

Variance, Var(X)
This measures the "spread" of the distribution. A larger variance means the results are more unpredictable. $$Var(X) = np(1-p)$$

(Syllabus note: The proofs for these formulas are not required. You just need to know them and how to use them!)

Example:

If you flip a fair coin 100 times (n=100, p=0.5), what are the expected number of heads and the variance?

Mean: $$E(X) = np = 100 \times 0.5 = 50$$ This makes perfect sense! You would expect to get 50 heads.

Variance: $$Var(X) = np(1-p) = 100 \times 0.5 \times (1-0.5) = 100 \times 0.5 \times 0.5 = 25$$

Did You Know?

The variance of a binomial distribution is at its highest when p = 0.5. This means an experiment is most unpredictable when success and failure are equally likely, like a coin flip!

Chapter Summary: Quick Review

Here are the absolute key points you need to remember for the Binomial Distribution.

  • Bernoulli Trial: A single experiment with two outcomes (Success/Failure).
  • Binomial Conditions (B.I.N.S.):
    • Binary outcomes
    • Independent trials
    • Number of trials is fixed
    • Same probability of success
  • Notation:
    $$X \sim B(n, p)$$
  • Probability Formula:
    $$P(X=k) = C_k^n p^k (1-p)^{n-k}$$
  • Mean (Expected Value):
    $$E(X) = np$$
  • Variance:
    $$Var(X) = np(1-p)$$

And that's it! By understanding these core ideas, you have a powerful tool for solving a huge range of probability problems. Practice using the B.I.N.S. checklist and the formula, and you'll master this topic in no time. Good luck!