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Placement Test Practice — Probability & Combinatorics
This practice test covers counting principles, permutations, combinations, the binomial theorem, basic probability, conditional probability, Bayes' theorem, and expected value. Apply counting techniques to compute probabilities and use Bayes' theorem for conditional reasoning.
Placement Test Practice — Probability & Combinatorics
Practice Test — 20 Questions
1. Compute $\binom{7}{2}$.
2. Compute $P(6,3)$.
3. How many 2-person committees from 8 people?
4. How many ways to arrange 4 books on a shelf?
5. A fair die is rolled. $P(\text{odd})$?
6. A card is drawn randomly. $P(\text{king})$?
7. $P(\text{not a king})$?
8. Two fair coins. $P(\text{exactly one head})$?
9. If $P(A)=0.4$ and $P(B)=0.3$, $A,B$ independent. $P(A \cap B)$?
10. $P(A \cup B)$ for the same events?
11. Bag: 3 red, 7 blue. Draw 2 without replacement. $P(\text{both red})$?
12. Expand $(x+1)^3$.
13. Pascal's: $\binom{6}{3}$ using the triangle.
14. Roll two dice. $P(\text{sum} = 11)$?
15. $E(X)$: win \$10 with prob $0.2$, lose \$2 with prob $0.8$.
16. $P(A|B) = 0.6$, $P(B) = 0.5$. Find $P(A \cap B)$.
17. How many 5-digit zip codes (digits 0–9, repeats okay)?
18. $\binom{10}{10}$?
19. A jar has 5 red, 5 green. $P(\text{red then green without replacement})$?
20. What is the sample space for flipping 2 coins?
Show Answer Key
1. $21$
2. $120$
3. $\binom{8}{2} = 28$
4. $4! = 24$
5. $3/6 = 1/2$
6. $4/52 = 1/13$
7. $48/52 = 12/13$
8. $2/4 = 1/2$
9. $0.12$
10. $0.4 + 0.3 - 0.12 = 0.58$
11. $\frac{3}{10} \cdot \frac{2}{9} = \frac{6}{90} = 1/15$
12. $x^3 + 3x^2 + 3x + 1$
13. $20$
14. $(5,6),(6,5) \to 2/36 = 1/18$
15. $10(0.2) + (-2)(0.8) = 2 - 1.6 = 0.40$
16. $0.6 \times 0.5 = 0.30$
17. $10^5 = 100{,}000$
18. $1$
19. $\frac{5}{10} \cdot \frac{5}{9} = 25/90 = 5/18$
20. $\{HH, HT, TH, TT\}$