Binomial Distribution
In short: Binomial distribution gives the probability of exactly k successes in n independent trials that each have the same success probability p and only two possible outcomes.
AP StatisticsProbabilityAP Statistics Unit 2
What is Binomial Distribution?
A binomial distribution counts successes in a fixed number of repeated tries. Think of guessing on 8 multiple-choice questions and counting how many you get right.
Four conditions must hold: a fixed number of trials \(n\); each trial is a success or failure; trials are independent; and the success probability \(p\) is the same every time. Then
\[P(X = k) = \binom{n}{k}\,p^k\,(1-p)^{n-k}\]
with mean \(\mu = np\) and standard deviation \(\sigma = \sqrt{np(1-p)}\).
How to spot it
- A problem repeats the same simple trial a fixed number of times and counts the successes.
- Each attempt is yes or no: a basket made or missed, a part defective or fine, a guess right or wrong.
- The question asks for the chance of exactly, at least or at most a certain number of successes.
- Sampling is done without replacement from a very large population, so independence is approximately true.
The method
- Check the four conditions and define a success clearly; identify \(n\) and \(p\).
- For an exact count, compute \(\binom{n}{k}p^k(1-p)^{n-k}\).
- For “at least” statements, use the complement when it is shorter, such as \(1 – P(X = 0)\).
- Find the mean \(np\) and standard deviation \(\sqrt{np(1-p)}\) to describe the typical count.
Binomial Distribution: solved example
Problem
A quiz has 8 multiple-choice questions with 4 options each. A student guesses every answer. Find the probability of exactly 3 correct answers, and the mean and standard deviation of the number correct.
Solution
Here \(n = 8\), \(p = 0.25\), \(k = 3\).
\(P(X = 3) = \binom{8}{3}(0.25)^3(0.75)^5 = 56 \times 0.015625 \times 0.2373 \approx\) 0.2076.
Mean: \(8(0.25) = 2\) correct. Standard deviation: \(\sqrt{8(0.25)(0.75)} = \sqrt{1.5} \approx 1.22\).
Try it
Question 1
Using the same quiz, find the probability of at least one correct answer.
Show the answer
\(1 – P(X = 0) = 1 – (0.75)^8 \approx 1 – 0.1001 = 0.8999\).
Question 2
A bakery’s loaves are underweight 5% of the time. In a batch of 20, find the expected number of underweight loaves.
Show the answer
\(np = 20(0.05) = 1\) loaf.
Binomial Distribution vs. Normal Distribution
The binomial distribution counts whole-number successes, so its bars are separate. The normal distribution is a smooth, continuous curve. When \(np\) and \(n(1-p)\) are both at least 10, the bars of the binomial distribution look close to a bell and the normal curve gives a good approximation.
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