IB Maths AI 4.5 Notes

Probability measures how likely an event is to happen. It is always a value between $0$ and $1$.

• If an event cannot happen, its probability is $0$.
• If an event is certain to happen, its probability is $1$.

A probability close to $0$ means the event is unlikely, while a probability close to $1$ means the event is likely.

There are four terms you are likely to come across: trial, outcome, sample space, and event.

• A trial is an experiment or process with uncertain results, such as tossing a coin or rolling a die.
• An outcome is one possible result of a trial. For example, when rolling a fair die, one possible outcome is $4$.
• The sample space is the set of all possible outcomes. It is often written as $U$.
  • For a single toss of a coin, $U=\{H,T\}$
  • For a single roll of a die, $U=\{1,2,3,4,5,6\}$
• An event is a set of outcomes from the sample space. For example, when rolling a die, the event of getting an even number is $A=\{2,4,6\}$.

If all outcomes in the sample space are equally likely, then:

$P(A)=\frac{n(A)}{n(U)}$

where $n(A)$ is the number of outcomes in event $A$ and $n(U)$ is the total number of outcomes in the sample space.

A fair six-sided die is rolled. Find the probability of rolling a $4$. 

There is $1$ favourable outcome and $6$ possible outcomes in total, so the probability of rolling a $4$ is $\frac{1}{6}$.