Probability

Dice: the probability scale runs from impossible (0) to certain (1).

Probability 概率 measures how likely an event 事件 is. It runs on a scale from $0$ (impossible) to $1$ (certain), and can be written as a fraction, decimal or percentage. We write $\text{P}(A)$ for the probability of event $A$.

Probability runs from $0$ (impossible) to $1$ (certain), with $\tfrac12$ an even chance.

For equally likely outcomes 结果,

The probability that an event does not happen is

Worked example. A bag has $3$ red and $5$ blue counters. Find the probability of not drawing red.

When outcomes are not equally likely, do an experiment. The relative frequency 相对频率 estimates the probability:

The more trials you do, the better the estimate. A fair 公平 object gives equal chances; one with bias 偏倚 does not; random 随机 means each outcome happens by chance.

The expected frequency 期望频数 is how many times you expect an event in $n$ trials:

Worked example. The probability of rolling a six is $\tfrac{1}{6}$. How many sixes are expected in $300$ rolls?

A fair spinner: eight identical sectors, so each number is an equally likely outcome.

For two or more events together (combined events 组合事件), two rules help:

• AND (both happen): multiply the probabilities — when the events are independent 独立事件 (one does not affect the other).
• OR (either happens): add the probabilities — when the events are mutually exclusive 互斥 (they cannot both happen).

Three pictures help you organise the work.

Sample space diagrams

A sample space diagram 样本空间图 is a table or grid listing every possible outcome — useful for two dice or two spinners. Count the outcomes you want out of the total.

A sample space diagram lists every outcome; for the total of two dice there are $36$ equally likely cells.

Venn diagrams

A Venn diagram 维恩图 sorts outcomes into overlapping sets. From it you can read $\text{P}(A \cap B)$ (in both) and $\text{P}(A \cup B)$ (in either).

The overlap is $\text{P}(A\cap B)=0.2$; everything inside either circle is $\text{P}(A\cup B)=0.6$; outside both is $0.4$.

Tree diagrams

A tree diagram 树状图 shows each stage as a set of branches. Write the probability on each branch and the outcome at the end. Multiply along the branches, then add the paths you want.

Worked example (with replacement). From the bag ($3$ red, $5$ blue), a counter is drawn, replaced, then a second is drawn. This is with replacement 有放回, so the chances do not change. Find the probability of two reds.

On a tree diagram, multiply the probabilities along the branches; the four outcomes' probabilities add to $1$.

Worked example (without replacement, Extended). Now the first counter is not replaced — drawing without replacement 无放回. After one red is taken, $2$ reds remain out of $7$:

Conditional probability 条件概率 is the probability of one event given that another has already happened. Read it from a Venn diagram, two-way table or tree diagram by looking only at the part that matches the condition.

Worked example. In a class of $30$, $18$ study French and, of those, $7$ also study German. A French student is picked. Find the probability they also study German.

Look only at the $18$ French students:

• Every probability lies between $0$ and $1$, and the probabilities of all the outcomes add to $1$.
• For "and" (both events) multiply the probabilities; for "or" (either event) add them. On a tree diagram, multiply along the branches.
• Watch for without replacement: the second probability changes because one item has already been removed.
• Expected frequency = probability × number of trials.