複合事象

Two events at once

• You roll a die and flip a coin. What's the chance of a six AND heads?
• Combined events need two rules: AND (multiply) and OR (add).

AND — multiply (independent 独立的 events)

• For two independent events (one doesn't affect the other):

OR — add (mutually exclusive 互斥 events)

• For two mutually exclusive events (they can't both happen):

Three ways to show outcomes

• Sample space 样本空间 diagram: a table of every outcome (two dice → $36$ cells).
• Venn diagram: sorts outcomes into overlapping sets.
• Tree diagram 树状图: a branch per stage; multiply along branches, then add the paths you want.

A Venn diagram sorts outcomes: the overlap is $\text{P}(A\text{ and }B)$, and everything inside either circle is $\text{P}(A\text{ or }B)$

With and without replacement 放回

• With replacement: chances stay the same each time.
• Without replacement (Extended): the total and the count change after each draw.
• Bag of $3$ red, $5$ blue: P(red, red) with replacement $= \dfrac{3}{8} \times \dfrac{3}{8} = \dfrac{9}{64}$. Without replacement $= \dfrac{3}{8} \times \dfrac{2}{7} = \dfrac{3}{28}$.

Venn diagrams show combined events: the overlap (intersection) is AND, the whole region (union) is OR.

Exactly one means two paths

• Draw twice from 2 red and 3 blue counters with replacement. Each stage has red $2/5$ and blue $3/5$. Exactly one red includes RB and BR, so $p=(2/5)(3/5)+(3/5)(2/5)=12/25$.
• At least one red is easier by complement: $p=1-(3/5)^2=16/25$. This includes RR as well as the two mixed paths; do not confuse exactly one with at least one.

You've got it