Probability basics and sample spacesIB MYP Maths Standard: Revision notes

Probability measures how likely an event is. It is a number from $0$ to $1$, written as a fraction, decimal or percentage.

• $0$: impossible (for example, rolling a 7 on a normal die)
• $\frac{1}{2}$: even chance
• $1$: certain The closer to $1$, the more likely the event. A probability can never be less than $0$ or greater than $1$.

When all outcomes are equally likely (a fair die, a fair coin, a counter taken at random): $P(\text{event})=\frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}.$ Worked example: a bag has 5 red, 3 blue and 2 green counters. $P(\text{blue})=\frac{3}{10}$, because 3 of the 10 counters are blue. Simplify fractions where possible.

The sample space is the list of all possible outcomes. List outcomes systematically so none are missed, or use a table (a sample space diagram) for two events. Two four-sided spinners give $4\times4=16$ outcomes: $(1,1),(1,2),\dots,(4,4)$. The total is $5$ for $(1,4)$, $(2,3)$, $(3,2)$, $(4,1)$, so $P(\text{total }5)=\frac{4}{16}=\frac{1}{4}$. A coin and a three-section spinner give $H1,H2,H3,T1,T2,T3$: $2\times3=6$ outcomes. For two dice there are $6\times6=36$ outcomes.

Either an event happens or it does not, so the probabilities add to $1$: $P(\text{not }A)=1-P(A).$ Example: $P(\text{red})=\frac{5}{10}$, so $P(\text{not red})=1-\frac{5}{10}=\frac{1}{2}$. This is often quicker than counting all the other outcomes.

Events are mutually exclusive if they cannot happen at the same time. For mutually exclusive events $A$ and $B$: $P(A\text{ or }B)=P(A)+P(B).$ Example: a counter cannot be both red and green, so $P(\text{red or green})=\frac{5}{10}+\frac{2}{10}=\frac{7}{10}$. If the events can both happen (for example 'an odd number' and 'a prime number' on a die), you cannot simply add. If a list of mutually exclusive events covers every outcome, their probabilities add to $1$.