6. Control questions and tasks / Контрольні запитання і завдання #

Counting principles #

1. State the product rule and the sum rule. Which word — “and” or “or” — signals each?
2. A choice of $k$ objects from $n$ is classified by two questions. Which two? Name the formula in each of the four cases.
3. When does order matter? Contrast Задача 1 (plates) with Задача 4 (balls in boxes): why is one $n^k$ and the other a multiset?

Arrangements, combinations, multisets #

4. Write $P(n,k)$ and $C(n,k)$ and explain the relation $P(n,k)=C(n,k)\cdot k!$ in words.
5. How many three-letter “words” can be formed from an alphabet of $8$ letters (a) if letters may repeat, (b) if they may not?
6. Explain the stars-and-bars picture: why does distributing $k$ identical balls into $n$ boxes give $\binom{n+k-1}{k}$?
7. In how many ways can a committee of $3$ be chosen from $10$ people? How does the answer change if the three take distinct roles (chair, secretary, treasurer)?

Probability #

8. State the classical (Laplace) definition of probability. What must be true of the outcomes for it to apply?
9. State the complement rule and explain why it is the natural tool for “at least one” problems (as in Задача 9).
10. When are two events independent, and what is $P(A\cap B)$ then? Why are the two draws in Задача 6 independent but the two draws in Задача 7 not?
11. Define conditional probability $P(A\mid B)$. Show how Задача 8 is the multiplication rule $P(\text{good})\,P(\text{1st}\mid\text{good})$.
12. A box has $6$ white and $4$ black balls; two are drawn without replacement. Find the probability that (a) both are white, (b) at least one is black.

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