📝 Chapter 5: Discrete Probability Distributions

Assessment AS Learning — Practice Quiz
🔄 Not Graded — Unlimited Retakes
Purpose: Self-check your skills with binomial, geometric, and hypergeometric distributions and their identification.
Score: 0 / 12
Topic 5.1 — Binomial Distribution
Question 1
A coin is tossed 10 times. \( P(\text{exactly 7 heads}) \) = ? (Decimal to 4 places)


Solution:
\( P(X=7) = \binom{10}{7}(0.5)^{10} = 120 \cdot \dfrac{1}{1024} = 0.1172 \).
Question 2
For \( X \sim \mathrm{Binomial}(n=20, p=0.3) \), find \( E(X) \) and \( \mathrm{Var}(X) \).
Solution:
\( E(X) = np = 20(0.3) = 6 \). \( \mathrm{Var}(X) = np(1-p) = 6(0.7) = 4.2 \).
Question 3
A student guesses on 8 multiple-choice questions (4 options each). \( P(\text{at least 1 correct}) \) = ? (Decimal to 4 places)


Solution:
\( P(\text{at least 1}) = 1 - P(0) = 1 - \binom{8}{0}(0.25)^0(0.75)^8 = 1 - (0.75)^8 \approx 1 - 0.1001 = 0.8999 \).
Topic 5.2 — Geometric Distribution
Question 4
A free-throw shooter has 80% accuracy. What is the probability that her first miss occurs on the 3rd attempt?


Solution:
"Success" = miss, p=0.2. \( P(X=3) = (0.8)^2 \cdot 0.2 = 0.64 \cdot 0.2 = 0.128 \).
Question 5
A die is rolled until a 6 appears. The expected number of rolls is:


Solution:
Geometric mean: \( \mu = \dfrac{1}{p} = \dfrac{1}{1/6} = 6 \) rolls.
Question 6
A salesperson succeeds on 30% of calls. \( P(\text{first success on 4th call}) \) = ? (Decimal to 4 places)


Solution:
\( P(X=4) = (0.7)^3 \cdot 0.3 = 0.343 \cdot 0.3 = 0.1029 \).
Topic 5.3 — Hypergeometric Distribution
Question 7
A batch of 50 items contains 8 defective. A sample of 10 is selected. \( P(\text{exactly 2 defective}) \) = ? (Decimal to 4 places)


Solution:
\( P = \dfrac{\binom{8}{2}\binom{42}{8}}{\binom{50}{10}} = \dfrac{28 \cdot 118\,030\,185}{10\,272\,278\,170} \approx 0.2964 \).
Question 8
A bag has 10 marbles: 6 red, 4 blue. 3 are drawn without replacement. \( P(\text{exactly 2 red}) \) = ? (Decimal to 4 places)


Solution:
\( P = \dfrac{\binom{6}{2}\binom{4}{1}}{\binom{10}{3}} = \dfrac{15 \cdot 4}{120} = \dfrac{60}{120} = 0.5 \).
Topic 5.4 — Choosing the Right Distribution
Question 9
"A factory sells 100 widgets in lots; 5 are defective. A sample of 8 is drawn from a lot." Which distribution applies?
Solution:
Sampling without replacement from a finite, mixed population → hypergeometric.
Question 10
"A coin is flipped repeatedly until tails appears." Which distribution describes the number of flips required?
Solution:
Counting trials until first success → geometric.
Question 11
"In a fixed sample of 30 voters surveyed by phone, asking each independently 'Do you support candidate A?'." Which distribution?
Solution:
Fixed n, independent trials, fixed p → binomial.
Question 12
A baseball player hits 0.300. What is the probability he gets his first hit on his 4th at-bat? (Decimal to 4 places)


Solution:
Geometric: \( P(X=4) = (0.7)^3 (0.3) = 0.343 \cdot 0.3 = 0.1029 \).