AP Statistics Binomial and Geometric Distributions Worksheet Solutions (PDF)

AP Statistics math worksheet solutions: Binomial and Geometric Distributions – Maths PDF

Cover of the book AP Statistics Math Worksheets with Answers (PDF)

Download as a PDFThe AP Statistics worksheets book with answersPrintable PDFSee the book ›

Sommaire

This binomial geometric answers 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.

Each solution states the distribution with its parameters, then shows the formula or calculator command and a rounded answer. Probabilities are rounded to four decimals, which matches calculator output.

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

Binomial geometric answers: exercises 1 to 10

Exercise 1 – Check BINS

(a) Yes. (b) No: the number of trials is not fixed (it is geometric-like, and cards are drawn without replacement). (c) Yes: the sample is far less than 10% of the output. (d) No: the trials are not independent.

(a) yes; (b) no; (c) yes; (d) no.

Exercise 2 – Binomial formula

\(P(X=3)=\frac{4!}{3!\,1!}(0.7)^{3}(0.3)^{1}=4(0.343)(0.3)=0.4116\).

0.4116.

Exercise 3 – binompdf

Binomial with n = 8 and p = 0.25: skewed right, mean 2.

X is binomial with n = 8 and p = 0.25.

\(P(X=2)=\text{binompdf}(8,0.25,2)=0.3115\). \(P(X=5)=0.0231\).

0.3115 and 0.0231.

Exercise 4 – binomcdf

\(P(X\le 3)=\text{binomcdf}(8,0.25,3)=0.8862\).

0.8862.

Exercise 5 – At least

Binomial with n = 8 and p = 0.25: skewed right, mean 2.

\(P(X\ge 4)=1-P(X\le 3)=1-0.8862=0.1138\).

0.1138.

Exercise 6 – Between

\(P(2\le X\le 4)=P(X\le 4)-P(X\le 1)=0.9727-0.3671=0.6056\).

0.6056.

Exercise 7 – Mean and standard deviation

\(\mu=np=25(0.8)=20\). \(\sigma=\sqrt{25(0.8)(0.2)}=\sqrt{4}=2\).

Mean 20, standard deviation 2.

Exercise 8 – Interpret the mean

The mean is a long-run average.

Over many sets of 25 attempts, she makes an average of 20 shots per set.

Exercise 9 – Shape

With p = 0.5 the distribution is symmetric and mound-shaped, centered at 5.

Symmetric; with p = 0.1 it is skewed right with its peak at 1.

Exercise 10 – Unusual result

\(P(X=10)=0.7^{10}=0.0282\).

0.0282, which is less than 5%, so making all ten is unusual.

Exercises 11 and up

Exercise 11 – 10 percent condition

The sample (4) is more than 10% of the population (30), and the probability changes after each draw.

No: the 10% condition fails (4 is about 13% of 30).

Exercise 12 – Geometric setting

(a) Yes. (b) No: fixed number of trials (binomial). (c) No: the probability changes, so trials are not independent.

(a) geometric; (b) binomial; (c) neither.

Exercise 13 – Geometric probability

Geometric with p = 0.2: always skewed right, with the highest bar at 1.

\(P(X=4)=(0.8)^{3}(0.2)=0.1024\).

0.1024.

Exercise 14 – More than n trials

More than 5 boxes means the first 5 are failures: \(0.8^{5}=0.3277\).

0.3277.

Exercise 15 – Geometric cumulative

\(P(X\le 3)=1-0.8^{3}=1-0.512=0.488\).

0.488.

Exercise 16 – Expected trials

\(\mu=\frac{1}{0.125}=8\).

You expect about 8 tests until the first pass; in the long run, it takes an average of 8 trials.

Exercise 17 – Geometric sd

\(\sigma=\frac{\sqrt{1-p}}{p}=\frac{\sqrt{0.8}}{0.2}\approx 4.472\).

About 4.47 boxes.

Exercise 18 – Exactly and at least

\(P(X=0)=0.95^{20}=0.3585\). \(P(X\ge 1)=1-0.3585=0.6415\).

0.3585 and 0.6415.

Exercise 19 – Compare binomial and geometric

(a) \(\text{binompdf}(5,0.6,3)=0.3456\). (b) \(0.4(0.6)=0.24\).

0.3456 and 0.24.

Exercise 20 – Multi-step

\(\mu=12(0.04)=0.48\). \(\sigma=\sqrt{12(0.04)(0.96)}=0.679\).

\(P(X > 2)=1-\text{binomcdf}(12,0.04,2)=0.0107\).

Mean 0.48; sd 0.679; P = 0.0107.

Key facts behind this binomial geometric answers

Key facts

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.

How to use this binomial geometric answers

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

Download the binomial geometric answers as a PDF with the button below and keep it with your notes.

All AP Statistics chapters: AP Statistics math.

Voter.. post

Free printable PDF