Welcome to the Binomial Distribution!
Have you ever wondered what the chances are of guessing correctly on a multiple-choice quiz, flipping 7 heads out of 10 coin tosses, or scoring at least 3 penalties in a football shootout? In statistics, whenever you repeat an experiment multiple times and each trial has only two possible outcomes (like win/lose, heads/tails, or success/failure), you are dealing with the Binomial Distribution.
Don't worry if this seems a bit overwhelming at first! In this chapter, we will break everything down step-by-step so that you can master every question type with confidence.
1. When Can We Use a Binomial Distribution?
Before using any formulas, you need to check whether a situation qualifies as a binomial experiment. A situation follows a binomial distribution if and only if it satisfies four key conditions.
A great way to remember these is the simple mnemonic BINS:
• B – Binary outcomes: There are only two possible outcomes for each trial. We label them success and failure.
• I – Independent trials: The outcome of one trial does not affect the outcome of any other trial.
• N – Number of trials is fixed: There is a predetermined, set number of trials (\(n\)).
• S – Same probability: The probability of success (\(p\)) is constant for every single trial.
Mathematical Notation
If a discrete random variable \(X\) follows a binomial distribution, we write:
\(X \sim \text{B}(n, p)\)
• \(X\) is the random variable (the number of successful outcomes we are counting).
• \(n\) is the total number of independent trials.
• \(p\) is the probability of success on a single trial.
• \(q = 1 - p\) is the probability of failure on a single trial.
Did you know? The prefix "Bi-" means two (like in bicycle or bilingual). That is your clue that there are only two outcomes per trial: success or failure!
Key Takeaway: Always check for BINS before applying any binomial formulas. If the probability changes between trials (for example, choosing cards from a deck without replacement), it is not binomial!
2. The Binomial Probability Formula
To find the probability of getting exactly \(r\) successes out of \(n\) trials, we use the Binomial Formula:
\(P(X = r) = {^n\text{C}_r} \times p^r \times (1 - p)^{n - r}\)
or writing \(q = 1 - p\):
\(P(X = r) = {^n\text{C}_r} \times p^r \times q^{n - r}\)
Breaking Down the Three Parts of the Formula
1. \({^n\text{C}_r}\) (Combinations): The number of different ways to choose \(r\) successes from \(n\) trials. You can calculate this directly on your scientific calculator using the \(\text{nCr}\) button.
2. \(p^r\): The probability of getting \(r\) successes multiplied together.
3. \(q^{n - r}\): The probability of getting the remaining \((n - r)\) failures multiplied together.
Calculator Tip
To compute \({^5\text{C}_2}\) on most calculators: Press 5 \(\rightarrow\) SHIFT \(\rightarrow\) \(\div\) (or the \(\text{nCr}\) key) \(\rightarrow\) 2 \(\rightarrow\) =. You should get \(10\).
Worked Example 1: Finding an Exact Probability
Question: A fair six-sided die is rolled \(8\) times. Find the probability of rolling exactly three \(6\)s.
Step-by-step Solution:
• Step 1: Identify \(n\), \(p\), and \(q\).
Number of rolls: \(n = 8\)
Success is rolling a \(6\): \(p = \frac{1}{6}\)
Failure is not rolling a \(6\): \(q = 1 - \frac{1}{6} = \frac{5}{6}\)
We want \(r = 3\) successes: \(X \sim \text{B}\left(8, \frac{1}{6}\right)\)
• Step 2: Set up the formula.
\(P(X = 3) = {^8\text{C}_3} \times \left(\frac{1}{6}\right)^3 \times \left(\frac{5}{6}\right)^{8 - 3}\)
\(P(X = 3) = {^8\text{C}_3} \times \left(\frac{1}{6}\right)^3 \times \left(\frac{5}{6}\right)^5\)
• Step 3: Calculate each component.
\({^8\text{C}_3} = 56\)
\(\left(\frac{1}{6}\right)^3 = \frac{1}{216}\)
\(\left(\frac{5}{6}\right)^5 = \frac{3125}{7776}\)
• Step 4: Multiply together.
\(P(X = 3) = 56 \times \frac{1}{216} \times \frac{3125}{7776} \approx 0.104\) (to 3 significant figures)
Key Takeaway: The powers in the formula must always add up to \(n\) (here, \(3 + 5 = 8\)). This is a quick way to check that you haven't made an index error!
3. Handling Inequalities: "At Least", "At Most", and "Fewer Than"
Exam questions often ask for the probability of a range of outcomes rather than just a single exact value. Translating the English words into mathematical statements is the secret to getting these right.
Translating Common Phrases
• "At least \(r\)" means \(X \ge r\) (e.g., at least \(2\) means \(X = 2, 3, 4, \dots\))
• "At most \(r\)" means \(X \le r\) (e.g., at most \(2\) means \(X = 0, 1, 2\))
• "Fewer than \(r\)" / "Less than \(r\)" means \(X < r\) (e.g., fewer than \(2\) means \(X = 0, 1\))
• "More than \(r\)" means \(X > r\) (e.g., more than \(2\) means \(X = 3, 4, 5, \dots\))
The "At Least One" Shortcut
A very common exam question asks for the probability of at least one success: \(P(X \ge 1)\).
Instead of calculating \(P(X = 1) + P(X = 2) + \dots + P(X = n)\), use the complement rule:
\(P(X \ge 1) = 1 - P(X = 0)\)
Since \(P(X = 0) = {^n\text{C}_0} \times p^0 \times q^n = 1 \times 1 \times q^n = q^n\), this simplifies neatly to:
\(P(X \ge 1) = 1 - q^n\)
Worked Example 2: Cumulative Probabilities
Question: A biased coin has a probability of landing on Heads of \(0.7\). The coin is flipped \(5\) times. Find the probability of getting:
(a) At most \(1\) head
(b) At least \(4\) heads
Step-by-step Solution:
Here, \(X \sim \text{B}(5, 0.7)\), so \(n = 5\), \(p = 0.7\), and \(q = 1 - 0.7 = 0.3\).
Part (a): At most 1 head \(\implies P(X \le 1)\)
\(P(X \le 1) = P(X = 0) + P(X = 1)\)
\(P(X = 0) = {^5\text{C}_0} \times (0.7)^0 \times (0.3)^5 = 1 \times 1 \times 0.00243 = 0.00243\)
\(P(X = 1) = {^5\text{C}_1} \times (0.7)^1 \times (0.3)^4 = 5 \times 0.7 \times 0.0081 = 0.02835\)
\(P(X \le 1) = 0.00243 + 0.02835 = 0.03078 \approx 0.0308\) (3 s.f.)
Part (b): At least 4 heads \(\implies P(X \ge 4)\)
\(P(X \ge 4) = P(X = 4) + P(X = 5)\)
\(P(X = 4) = {^5\text{C}_4} \times (0.7)^4 \times (0.3)^1 = 5 \times 0.2401 \times 0.3 = 0.36015\)
\(P(X = 5) = {^5\text{C}_5} \times (0.7)^5 \times (0.3)^0 = 1 \times 0.16807 \times 1 = 0.16807\)
\(P(X \ge 4) = 0.36015 + 0.16807 = 0.52822 \approx 0.528\) (3 s.f.)
Key Takeaway: Always list out the integer values included in the inequality before calculating so you don't accidentally miss a term (like \(X = 0\)).
4. Mean (Expected Value) of a Binomial Distribution
The mean or expected value, written as \(E(X)\) or \(\mu\), represents the average number of successes you would expect if you repeated the experiment many times.
Formula for the Mean
\(E(X) = \mu = np\)
Why does this make intuitive sense?
Imagine you roll a fair six-sided die \(60\) times. How many times would you expect to get a \(6\)?
Since the probability on each roll is \(\frac{1}{6}\), you would naturally expect:
\(E(X) = 60 \times \frac{1}{6} = 10\text{ times}\)
Worked Example 3: Expected Value
Question: In a manufacturing plant, \(4\%\) of items produced are defective. A quality control inspector takes a random sample of \(250\) items. Find the expected number of defective items in the sample.
Solution:
• Number of trials: \(n = 250\)
• Probability of defect (success in this context): \(p = 0.04\)
• \(E(X) = np = 250 \times 0.04 = 10\)
The expected number of defective items is \(10\).
Key Takeaway: The mean is simply \(n \times p\). It gives you the long-term average outcome.
5. Common Mistakes to Avoid
• Forgetting \(P(X = 0)\): When working out "at most \(2\)", remember it means \(P(X = 0) + P(X = 1) + P(X = 2)\). Many students forget the zero outcome!
• Mixing up \(p\) and \(q\): Make sure the power on \(p\) matches the number of successes (\(r\)) and the power on \(q\) matches the number of failures (\(n - r\)).
• Confusing "fewer than" and "at most": "Fewer than \(3\)" means \(X \le 2\) (does NOT include \(3\)), whereas "at most \(3\)" means \(X \le 3\) (DOES include \(3\)).
• Rounding too early: Keep full precision in your intermediate steps on your calculator and round only at the very end (usually to 3 significant figures).
6. Chapter Summary & Quick Review
• Binomial conditions (BINS): Binary outcomes, Independent trials, Number of trials fixed, Same probability of success.
• Notation: \(X \sim \text{B}(n, p)\)
• Formula: \(P(X = r) = {^n\text{C}_r} \times p^r \times q^{n - r}\), where \(q = 1 - p\)
• At least one trick: \(P(X \ge 1) = 1 - P(X = 0) = 1 - q^n\)
• Mean / Expected Value: \(E(X) = np\)