Key takeaway: The binomial distribution gives the chance of exactly k successes when a process is repeated n times , each repetition independent and each with the same success probability p.
Grade 12 – AP StatisticsAP Statistics Unit 2Probability and Counting
Definition of Binomial Distribution
The binomial distribution models counts of successes. A situation qualifies when four conditions are met: the number of attempts is set in advance, every attempt ends as a success or a failure, the attempts do not influence each other, and the chance of success stays constant.
The random variable \(X\) counts the successes. Shooting 8 free throws at a steady 75% rate is a typical case. The shape is symmetric when \(p=0.5\) and leans away from whichever outcome is more likely otherwise.
Formula and notation
If \(X\) is binomial with \(n\) attempts and success probability \(p\):
\[ P(X=k)=\binom\,{n}{k}\,(1-p)^{n-k}\,p^{k},\qquad \mu=np,\qquad \sigma=\sqrt{np(1-p)}. \]
Short form: \(X\sim B(n,p)\). The binomial coefficient counts the arrangements of the successes.
Key facts
- All four conditions must hold; sampling without replacement from a small population breaks independence.
- The mean number of successes is n times p, which need not be a whole number.
- The coefficient n choose k counts how many orders give exactly k successes.
- Cumulative probabilities such as at most k add the individual terms.
Binomial Distribution example
A player makes 75% of her free throws. In a game she takes 8, independently. Find the probability she makes exactly 6, and the mean and standard deviation of the makes.
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Check the setting: \(n=8\), \(p=0.75\), independent shots.
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\(P(X=6)=28\cdot(0.25)^2\cdot(0.75)^6\approx 28\cdot 0.0625\cdot 0.17798\approx 0.3115\), where \(\binom\,{8}{6}=28\).
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Mean: \(\mu=8(0.75)=6\). Standard deviation: \(\sigma=\sqrt{8(0.75)(0.25)}=\sqrt{1.5}\approx 1.22\).
The probability of exactly six makes is about 0.3115, with mean 6 and standard deviation near 1.22.
Common mistake
Watch out
A student calculates \(P(X=6)=(0.75)^6(0.25)^2\approx 0.0111\) and stops.
Why it is wrong
That product is the probability of one specific order, such as six makes followed by two misses. The six makes can fall in any of 28 positions, so the product must be multiplied by 28, giving 0.3115. The binomial distribution always includes that counting factor.
Related terms and lessons
Browse the glossary: A–Z list · AP Statistics vocabulary · Probability and Counting