How Do You Calculate the Probability of an Event?

To calculate the probability of an event with equally likely outcomes, divide the number of outcomes in the event by the total number of possible outcomes:

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

Here, $A$ is the event, $U$ is the sample space, $n(A)$ is the number of favourable outcomes, and $n(U)$ is the total number of possible outcomes.

The Reasoning

A trial is an action with an uncertain result, such as rolling a die. An outcome is one possible result, while an event contains one or more outcomes of interest.

Suppose a bag contains 5 red, 3 blue, and 2 green counters. One counter is selected randomly, so each counter is equally likely to be chosen. For the event $A$, “selecting a red or green counter”:

$$n(A)=5+2=7, \qquad n(U)=5+3+2=10$$

Therefore,

$$P(A)=\frac{7}{10}=0.7$$

A probability must satisfy $0\leq P(A)\leq 1$. A probability of $0$ represents an impossible event, while $1$ represents a certain event.

The counting formula applies only when outcomes are equally likely. Otherwise, calculate the event probability by adding the probabilities of its individual outcomes rather than counting outcomes alone.

For the example, the probability of not selecting red or green is $1-0.7=0.3$.

Experimental probability estimates probability from observed data. If red or green appears 68 times in 100 repeated selections with replacement, the experimental probability is $68/100=0.68$. It need not equal the theoretical value exactly, but should generally approach it as the number of trials increases.

A common misconception is dividing by the number of favourable outcomes. The denominator must be the total number of equally likely outcomes in the sample space.

IB Exam Technique

This is SL 4.5: Trial and outcome, typically tested as a Paper 1 short-response skill, although it may appear within longer Paper 2 questions. For “calculate” or “find,” show the probability fraction before giving the decimal, and check that your final answer lies between $0$ and $1$.