Introduction to Probability

All the ways it could go

• The sample space 样本空间 is the set of all possible outcomes of a chance process.
• Each individual result is an outcome 结果; an event is a set of outcomes.
• Rolling a die: sample space $\{1,2,3,4,5,6\}$; the event "even" $= \{2,4,6\}$.
• Listing the sample space is the first step in finding a probability.

The basic rules

• Every probability satisfies $0 \le P(A) \le 1$ (never negative, never above $1$).
• The probabilities of all outcomes in the sample space sum to $1$.
• Complement rule 补集法则: $P(A^c) = 1 - P(A)$ — "not $A$" is whatever's left.
• These rules constrain every valid probability model.

Probability runs from 0 (impossible) to 1 (certain)

Equally likely outcomes

• When outcomes are equally likely 等可能, probability is just counting.
• $P(A) = \dfrac{\text{number of outcomes in } A}{\text{total number of outcomes}}$.
• $P(\text{even on a die}) = 3/6 = 0.5$.
• This "favorable over total" rule works only when outcomes are equally likely.

Picturing events

• A Venn diagram 韦恩图 draws events as overlapping circles inside the sample space.
• The overlap is the intersection (both events); the whole shaded region is the union (either).
• A two-way table does the same job with counts in rows and columns.
• Both make "and," "or," and "not" easy to see and compute.

Write from a probability of 0.2

• A spinner has 5 equal sectors.
• One sector is marked win. The lesson probabilities 0.5 and 0.75 are other chance setups.