AP Statistics Binomial and Geometric Distributions Lesson (Free PDF)
This binomial geometric lesson for AP Statistics covers binomial and geometric distributions (Common Core AP STAT 4.10, AP STAT 4.11, AP STAT 4.12, HSS-MD.A.3, HSS-CP.B.9), with a free PDF to print.
A binomial random variable counts successes in a fixed number n of independent trials with the same success probability p; use the BINS conditions to check. Its mean is np. A geometric random variable counts the trials until the first success, with P(X = n) = (1-p)^(n-1) p and mean 1/p.
AP Statistics · Common Core State Standards: AP STAT 4.10, AP STAT 4.11, AP STAT 4.12, HSS-MD.A.3, HSS-CP.B.9
Key facts in this binomial geometric lesson
- Binomial conditions: Binary, Independent, fixed Number, Same probability.
- Binomial: mean = np, standard deviation = sqrt(np(1-p)); use binompdf and binomcdf.
- At least k successes: 1 – binomcdf(n, p, k – 1).
- Geometric: trials until the first success; P(X = n) = (1-p)^(n-1) p; mean 1/p.
- Without replacement, treat trials as independent only if n is at most 10% of the population.
What is a binomial distribution?
A binomial setting has a fixed number \(n\) of independent trials, each with two outcomes (success or failure) and the same probability of success \(p\). The random variable \(X\) counts the successes. Check the conditions with BINS.
Example: a student guesses on \(n=8\) multiple-choice questions with 4 options. Then \(p=0.25\), and \(X\) is the number of correct guesses.
\[P(X=k)=\frac{n!}{k!\,(n-k)!}\,p^{k}(1-p)^{n-k}\]
For exactly 2 correct: \(P(X=2)=\frac{8!}{2!\,6!}(0.25)^{2}(0.75)^{6}=28(0.0625)(0.17798)=0.3115\).
The mean and standard deviation of a binomial are \(\mu_X=np\) and \(\sigma_X=\sqrt{np(1-p)}\). Here \(\mu_X=8(0.25)=2\) and \(\sigma_X=\sqrt{8(0.25)(0.75)}=1.225\).
How do you compute binomial probabilities?
- State the distribution: X is binomial with n = 8 and p = 0.25.
- Write the event, such as \(P(X\ge 3)\).
- Use binompdf for one value and binomcdf for “at most.” For “at least,” use the complement.
- Answer in context.
\(P(X\ge 3)=1-P(X\le 2)=1-0.6785=0.3215\).
Interpreting a binomial answer matters as much as computing it. If the chance of 4 or more correct guesses is about 0.114, then a student who guesses on every question will reach that score on roughly one quiz in nine. A result with probability below 5% is usually called unusual, and the count of successes should be described with the context, for example “the number of correct answers out of 8 questions.” The shape depends on \(p\): it is skewed right when \(p\) is small, symmetric when \(p=0.5\), and skewed left when \(p\) is large.
What is a geometric distribution?
A geometric setting has independent trials with the same probability of success \(p\), repeated until the first success. \(X\) counts the trials needed. There is no fixed \(n\).
\[P(X=n)=(1-p)^{n-1}\,p,\qquad \mu_X=\frac{1}{p}\]
If 20% of cereal boxes have a prize (\(p=0.2\)), the chance that the first prize appears in the 4th box is \(0.8^{3}(0.2)=0.1024\). You expect to open \(\frac{1}{0.2}=5\) boxes on average.
Binomial geometric lesson: questions and answers
What does BINS stand for?
BINS stands for Binary outcomes (success or failure), Independent trials, a fixed Number of trials, and the Same probability of success on each trial. All four must hold for a binomial model.
How do I find ‘at least’ probabilities?
Use the complement. For P(X >= 3) in n trials, compute 1 – P(X <= 2) = 1 - binomcdf(n, p, 2). Doing this with the calculator avoids adding many terms.
What is the difference between binomial and geometric?
A binomial counts successes in a fixed number of trials. A geometric counts the number of trials until the first success, so its number of trials is not fixed.
Can I use a binomial when sampling without replacement?
Yes if the sample is less than 10% of the population. Then the probability of success changes very little, so the trials are approximately independent.
Why is the geometric distribution always skewed right?
The probabilities start at p for the first trial and keep decreasing by a factor of (1 – p) each trial. The long tail extends to the right.
How to use this binomial geometric lesson
Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.
Download the binomial geometric lesson as a PDF with the button below and keep it with your notes.
Download the binomial geometric lesson as a PDF with the button below and keep it with your notes.
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