Events (Probability)

Probability of events

The Probability of events ranges between 0 and 1, and it measures how likely it is that an event will happen. If the probability of an event is 0 (zero), it is considered impossible. If the probability of an event is 1, it is certain that it will happen. If the probability of an event is 0.5, then the event is equally likely to happen as it is not likely to happen. Any event with a probability between 0 and 0.5 is considered unlikely to happen, and any event with a probability between 0.5 and 1 is considered likely to happen. Let's see this more clearly below.

The probability of events - StudySmarter Originals

Probabilities can be expressed in Fractions, decimals or percentages. For example, if an event has a probability of $\frac{1}{2}$, it is the same as saying 0.5 or 50%.

$Probabilityofanyevent=\frac{Numberofoutcomesthatsatisfyarequirement}{Totalnumberofpossibleoutcomes}$

What are independent events?

Two events (A and B) are independent if the fact that A has happened does not affect the probability of B happening, and vice versa. For example, when tossing a coin twice, the outcome of the first event does not affect the probability of the second. The probability of getting heads the first time is $\frac{1}{2}$, and the probability of getting heads the second time is also $\frac{1}{2}$, the probability does not change no matter how many times you toss the coin. The outcome of the previous event has no effect on the following one.

Independent events probability formula

When two events are independent, you can use the following multiplication rule :

$P(AandB)=P\left(A\right)\times P\left(B\right)$ using set Notation: $P(A\cap B)=P\left(A\right)\times P\left(B\right)$

This rule can be read as the probability of A and B happening together equals the probability of A times the probability of B.

What are dependent events?

Two events (A and B) are dependent if the fact that A has happened affects the probability of B happening and vice versa.

Dependent events probability formula

The multiplication rule for dependent events is as follows:

$P(AandB)=P\left(A\right)\times P(BafterA)$ using set Notation: $P(A\cap B)=P\left(A\right)xP\left(B\right|A)$

This rule can be read as the probability of A and B happening together equals the probability A times the probability of B after A occurred.

What are mutually exclusive events?

Mutually exclusive events have no outcomes in common. Therefore, they cannot occur together. For example, getting heads or tails when tossing a coin are mutually exclusive events, as you cannot get both at the same time.

Using a Venn diagram, mutually exclusive events can be represented as follows:

Venn Diagram of mutually exclusive events, Marilu García De Taylor - StudySmarter Originals

You can learn more about Venn Diagrams too.

Mutually exclusive events probability formula

In the case of mutually exclusive events, you can use the following addition rule to calculate the combined probabilities:

$P(AorB)=P\left(A\right)+P\left(B\right)$

This rule can be read as the probability of A or B happening equals the probability of A plus the probability of B.

In this case, the probability of A and B happening together is 0 (zero).

$P(AandB)=P(A\cap B)=0$

What are combined or compound events in probability?

Combined or compound events consist of two or more experiments being carried out together. When working with combined events, it is useful to visualise all the possible outcomes using a Tree Diagram.