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# 统计代写|How to count stat 代写

## 统计代考

Calculating the naive probability of an event $A$ involves counting the number of pebbles in $A$ and the number of pebbles in the sample space $S$. Often the sets we need to count are extremely large. This section introduces some fundamental methods for counting; further methods can be found in books on combinatorics, the branch of mathematics that studies counting.
1.4.1 Multiplication rule
In some problems, we can directly count the number of possibilities using a basic but versatile principle called the multiplication rule. We’ll see that the multiplication rule leads naturally to counting rules for sampling with replacement and sampling without replacement, two scenarios that often arise in probability and statistics.

Theorem 1.4.1 (Multiplication rule). Consider a compound experiment consisting of two sub-experiments, Experiment A and Experiment B. Suppose that Experiment A has $a$ possible outcomes, and for each of those outcomes Experiment B has $b$ possible outcomes. Then the compound experiment has $a b$ possible outcomes.
To see why the multiplication rule is true, imagine a tree diagram as in Figure 1.2. Let the tree branch $a$ ways according to the possibilities for Experiment A, and for each of those branches create $b$ further branches for Experiment B. Overall, there are $\underbrace{b+b+\cdots+b}_{a}=a b$ possibilities.
1.4.2. It is often easier to think about the experiments as being in chronological order, but there is no requirement in the multiplication rule that Experiment A has to be performed before Experiment $\mathrm{B}$.

Example 1.4.3 (Runners). Suppose that 10 people are running a race. Assume that ties are not possible and that all 10 will complete the race, so there will be welldefined first place, second place, and third place winners. How many possibilities are there for the first, second, and third place winners?

Solution: There are 10 possibilities for who gets first place, then once that is fixed there are 9 possibilities for who gets second place, and once these are both fixed there are 8 possibilities for third place. So by the multiplication rule, there are $10 \cdot 9 \cdot 8=720$ possibilities.

We did not have to consider the first place winner first. We could just as well have said that there are 10 possibilities for who got third place, then once that is fixed there are 9 possibilities for second place, and once those are both fixed there are 8 possibilities for first place. Or imagine that there are 3 platforms, which the first, second, and third place runners will stand on after the race. The platforms are gold, silver, and bronze, allocated to the first, second, and third place runners, respectively. Again there are $10 \cdot 9 \cdot 8=720$ possibilities for how the platforms will be occupied after the race, and there is no reason that the platforms must be considered in the order (gold, silver, bronze).

Example 1.4.4 (Chessboard). How many squares are there in an $8 \times 8$ chessboard, as in Figure $1.3 ?$ Even the name ” $8 \times 8$ chessboard” makes this easy: there are $8 \cdot 8=64$ squares on the board. The grid structure makes this clear, but we can also think of this as an example of the multiplication rule: to specify a square, we can specify which row and which column it is in. There are 8 choices of row, for each of which there are 8 choices of column.

Furthermore, we can see without doing any calculations that half the squares are white and half are black. Imagine rotating the chessboard 90 degrees clockwise. Then all the positions that had a white square now contain a black square, and vice versa, so the number of white squares must equal the number of black squares. We

## 统计代考

1.4.1 乘法规则

1.4.2.通常更容易将实验视为按时间顺序排列，但乘法规则中没有要求实验 A 必须在实验 $\mathrm{B}$ 之前执行。