The probability scale

Two labels do not mean equal chances

• A club has 10 raffle tickets: 3 win a prize and 7 do not. One ticket is chosen so that each ticket has the same chance. Winning and losing are two possible labels, but they are not equally likely.
• Probability 概率 measures how likely an event is. Winning has probability $\dfrac{3}{10}=0.3=30\%$. Count the equally likely tickets, rather than just the two labels.

Place a chance on the scale

• Probabilities lie from $0$ to $1$, including both ends. A probability of $0$ is impossible, $1$ is certain and $\dfrac{1}{2}=0.5=50\%$ is an even chance. Values below $0$ or above $1$ are invalid.
• For one roll of an ordinary six-sided die, a $7$ is impossible and a result from $1$ to $6$ is certain. A fair 公平 coin has equal chances of heads and tails; merely having two possible results does not guarantee an even chance.

Count equally likely outcomes 结果

• A bag contains 3 red and 5 blue counters. Assume each of the 8 counters has the same chance of being chosen. The favourable outcomes for red are the 3 red counters.
• Use the total number of counters in the denominator, not the number of colours or the number of blue counters. The fraction can be left exact or converted to a decimal or percentage.

Count sectors, not colour names

• An ideal fair spinner has 8 equal sectors: 4 blue, 3 red and 1 green. Each sector has probability $\dfrac{1}{8}$, so blue has probability $\dfrac{4}{8}=\dfrac{1}{2}$ and green has probability $\dfrac{1}{8}$.
• There are three colours, but each colour does not have probability $\dfrac{1}{3}$. Equal chances belong to the sectors in this model, and a colour can cover several sectors.

Read the whole total from a table

• This table records how each of 30 students travels to school. Choose one student at random, giving each student an equal chance. The chance they use the bus is $\dfrac{9}{30}=\dfrac{3}{10}$.
• Check what the table counts. Use the total number of students, not the four travel methods. Students who do not walk include all three other rows: $9+6+3=18$.

Not an event includes every other result

• The complement 补集 of an event means all outcomes where it does not happen. An event and its complement cover every possibility without overlap, so their probabilities add to $1$.
• A bag has 2 red, 3 blue and 5 green counters, each equally likely to be chosen. Not blue includes red and green: $\dfrac{2+5}{10}=\dfrac{7}{10}$. It does not mean only red.

Account for every possible outcome

• If a counter must be red, blue or green, those three colour probabilities add to $1$. If red has probability $0.35$ and blue has probability $0.25$, green must have probability $1-0.35-0.25=0.40$.
• Check a proposed model before using it. Three exhaustive colour probabilities of $0.4$, $0.45$ and $0.3$ add to $1.15$, so they cannot describe the bag. Each individual number is in range, but the total is wrong.

Read the notation: Extended

• $\text{P}(A)$ means the probability of event $A$. A letter names the event: for example, let $A$ mean “the chosen counter is blue”.
• $A'$ means “not $A$”, so $\text{P}(A')=1-\text{P}(A)$. If $\text{P}(A)=0.3$, then $\text{P}(A')=0.7$. Core questions can use words instead; this notation is not required at Core.

Show what you counted

• For a fresh problem, identify the possible outcomes and why they are equally likely. Count the favourable ones and the full total, then give a fraction, decimal or percentage.
• For “not”, include every other outcome or subtract from $1$. Check that the answer lies between $0$ and $1$ and that a complete list of separate outcomes has total probability $1$.

Keep the model and the total clear