Probability

Probability Formula

The formula to calculate the probability of an event is as follows:

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

Probabilities can be expressed in Fractions, decimals or percentages. For example, when tossing a coin, the probability of it landing on tails is $\frac{1}{2}$, which is the same as saying 0.5 or 50%.

Probability Rules

The main rules of probability that you need to keep in mind when calculating probabilities are:

• The probability of an event happening ranges between 0 (zero) and 1:

$0\le P\left(A\right)\le 1$

• The sum of the probabilities of all possible outcomes equals 1.

• Complement rule: The probability that an event does not happen equals 1 minus the probability of the event happening:

$P(notA)=1-P\left(A\right)$

• General addition rule: The probability of A or B happening equals the probability of A plus the probability of B minus the probability of A and B happening together:

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

• Addition rule for mutually exclusive events: Mutually exclusive events cannot happen at the same time. To calculate the probability of A or B happening in this case, we use the addition rule:

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

The rule changes because for mutually exclusive events, $P(AandB)=0$

• Multiplication rule for independent events: Two events are independent when the occurrence of one event does not affect the probability of another one happening. If A and B are independent, the probability of A and B happening together equals the probability of A times the probability of B:

$P(AandB)=P\left(A\right)\times P\left(B\right)$

• Conditional Probability: In this case, the probability of an event will be affected by another event that has happened. The Conditional Probability of B given that A happened is:

$P\left(B\right|A)=\frac{P(AandB)}{P\left(A\right)}$

The denominator will be the probability of the given event.

How do you represent probability?

When solving probability problems, it can get very confusing to work out all the possible outcomes from an experiment. To make this task easier, you can use diagrams specially designed for this purpose to create visual representations of all the possible outcomes. These diagrams are the Venn diagram and the Tree diagram. Let's see when you need to use each one.

Venn diagrams

Venn diagrams are very useful when solving probability problems, as they help you represent events graphically. A rectangle is used to represent the sample space (S), and inside the rectangle, you can draw oval shapes representing each event. You can also include the frequencies or the probabilities of each event in the diagram. Let's see the most common scenarios that you can represent with Venn diagrams for two events, A and B:

1. Event A and B: In this case, both A and B occur, represented by the intersection of the two ovals.

Venn diagram of the event A and B, Marilú García De Taylor - StudySmarter Originals

2. Event A or B: In this case, A or B or both occur, represented by the union of the two ovals.

Venn diagram of the event A or B, Marilú García De Taylor - StudySmarter Originals

3. Event not A: In this case, A does not occur, and it represents the complement of A.

Venn diagram of the event not A, Marilú García De Taylor - StudySmarter Originals

Please follow the link to the Venn Diagrams article to expand your knowledge about this topic.

Tree diagrams

Tree diagrams are especially useful to represent all the possible outcomes when you have two or more events happening one after the other. To create a tree diagram, draw a branch for each outcome in an event. Each branch should point to its corresponding outcome and include the probability of occurrence of each outcome.

You can read the Tree Diagram article to expand your knowledge about this topic.

Calculating probability

Here are a couple of examples of how to calculate probability:

For more examples, check out the Probability Calculations article.

What is conditional probability?

As mentioned in the Probability Rules, conditional probability refers to the probability of an event happening, given that another event has happened. The conditional probability of B given that A happened is:

$P\left(B\right|A)=\frac{P(AandB)}{P\left(A\right)}$

The denominator will be the probability of the given event.

Please follow the link to the conditional probability article to learn more about this topic.

What is a probability distribution?

A Probability Distribution is a table or equation that associates each possible outcome of a random variable with its corresponding probabilities. A random variable is a variable whose value is defined by the outcome of a random experiment. A variable is discrete when it can only take certain numerical values within a given interval. A variable is continuous when it can take infinite values within an interval.

Discrete Probability Distribution

A discrete Probability Distribution lists all the probabilities for each outcome of the random variable using a table.

Continuous probability distribution

The probability distribution of a continuous random variable is represented by an equation. This equation is called the probability density function, which has the following characteristics:

• $y=f\left(x\right)$

• $y\ge 0$ for all values of x

• The area under the curve of the function is equal to 1.

The Probability Distribution study set has more about this topic!