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Disjoint and Overlapping Events

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Sometimes you might be interested in investigating the probability of two events happening simultaneously. Is it possible? If it is, then do they have any outcomes in common? This is when disjoint and overlapping events come into play. In this article, we will explore their definition, formula, and some practical examples.

First of all, let's define what a compound event is.

A compound event of two events A and B is defined as the union of all the outcomes from both events A and B, or the intersection of the common outcomes shared by A and B.

If you want to work out the probability of A or B happening P(A or B), then you need to consider whether the two events have any outcomes in common or not.

What are Disjoint or Mutually Exclusive Events in Probability?

Disjoint or mutually exclusive events are events that 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, disjoint events can be represented as follows:

Venn diagram of disjoint events - StudySmarter Originals

Venn diagrams help you represent events graphically. A rectangle is used to represent the sample space (S), and inside the rectangle, you draw oval shapes representing each event. You can also include the frequencies or the probabilities of each event in the diagram.

Disjoint Events Probability Formula

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

$\mathbit{P}\mathbf{\left(}\mathbit{A}\mathbf{}\mathbit{o}\mathbit{r}\mathbf{}\mathbit{B}\mathbf{\right)}\mathbf{=}\mathbit{P}\mathbf{\left(}\mathbit{A}\mathbf{\right)}\mathbf{+}\mathbit{P}\mathbf{\left(}\mathbit{B}\mathbf{\right)}$

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).

$\mathbit{P}\mathbf{\left(}\mathbit{A}\mathbf{}\mathbit{a}\mathbit{n}\mathbit{d}\mathbf{}\mathbit{B}\mathbf{\right)}\mathbf{=}\mathbf{0}$

1. When tossing a coin, what is the probability of getting heads or tails?

A = coin landing on heads

B = coin landing on tails

Venn diagram of a disjoint event - Example 1, Marilú García De Taylor - StudySmarter Originals

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

$P\left(AorB\right)=\frac{1}{2}+\frac{1}{2}=1$

2. When rolling a die, what is the probability of getting a 3 or an even number?

A = getting a 3

B = getting an even number

The event of getting a 3 has only one favorable outcome, but getting an even number has 3 favorable outcomes, which are 2, 4, and 6.

Venn diagram of a disjoint event - Example 2, Marilú García De Taylor - StudySmarter Originals

$P\left(AorB\right)=P\left(A\right)+P\left(B\right)\phantom{\rule{0ex}{0ex}}=\frac{1}{6}+\frac{3}{6}=\frac{4}{6}=\frac{2}{3}=0.6667=66.67%$

What are Overlapping Events in Probability?

Overlapping events are compound events with one or more outcomes in common.

The Venn diagram that represents overlapping events is as follows:

Venn diagram of overlapping events - StudySmarter Originals

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

Overlapping Events Probability Formula

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:

$\mathbit{P}\mathbf{\left(}\mathbit{A}\mathbf{}\mathbit{o}\mathbit{r}\mathbf{}\mathbit{B}\mathbf{\right)}\mathbf{=}\mathbit{P}\mathbf{\left(}\mathbit{A}\mathbf{\right)}\mathbf{+}\mathbit{P}\mathbf{\left(}\mathbit{B}\mathbf{\right)}\mathbf{-}\mathbit{P}\mathbf{\left(}\mathbit{A}\mathbf{}\mathbit{a}\mathbit{n}\mathbit{d}\mathbf{}\mathbit{B}\mathbf{\right)}$

1. There are 15 students in a class, 6 students are studying French only, 4 are studying Spanish only, and 5 are studying both languages. What is the probability that a randomly selected student is studying French or Spanish only?

A = students studying French

B = students studying Spanish

The Venn Diagram below shows the number of students in each category.

Venn diagram of an overlapping event - Example 1 - StudySmarter Originals

$P\left(AorB\right)=P\left(A\right)+P\left(B\right)-P\left(AandB\right)\phantom{\rule{0ex}{0ex}}=\frac{6}{15}+\frac{4}{15}-\frac{5}{15}\phantom{\rule{0ex}{0ex}}=\frac{5}{15}=\frac{1}{3}=0.333=33.3%$

2. When rolling a die, what is the probability of getting a number less than 3 or an odd number?

A = getting a number less than 3

B = getting an odd number

The event of getting a number less than 3 has only 2 favorable outcomes: 1 and 2. Getting an odd number has 3 favorable outcomes, which are 1, 3, and 5. Outcome 1 is shared by both events.

Venn diagram of an overlapping event - Example 2 - StudySmarter Originals

$P\left(AorB\right)=P\left(A\right)+P\left(B\right)-P\left(AandB\right)\phantom{\rule{0ex}{0ex}}=\frac{2}{6}+\frac{3}{6}-\frac{1}{6}\phantom{\rule{0ex}{0ex}}=\frac{4}{6}=\frac{2}{3}=0.6667=66.67%$

Disjoint and Overlapping Events - Key takeaways

• A compound event of two events A and B is defined as the union of all the outcomes from both events A and B, or the intersection of the common outcomes shared by A and B.

• To work out the probability of A or B happening P(A or B), consider whether the two events have any outcomes in common or not.

• Disjoint or mutually exclusive events are events that cannot occur together.

• The disjoint events probability formula is: $P\left(AorB\right)=P\left(A\right)+P\left(B\right)$

• Overlapping events are compound events with one or more outcomes in common.

• The overlapping events probability formula is: $P\left(AorB\right)=P\left(A\right)+P\left(B\right)-P\left(AandB\right)$

Disjoint or mutually exclusive events are events that have no outcomes in common, therefore they cannot occur together. Overlapping events are compound events with one or more outcomes in common.

Events are disjoint if they have no outcomes in common. If two events have one or more outcomes in common, then they are overlapping events.

An example of an overlapping event is getting a number less than 3 or an even number when rolling a die. This is because both events share the outcome 2.

When we represent events using Venn Diagrams, oval shapes are used to represent each event and their possible outcomes. Each oval shape represents a set. If two sets do not have any outcomes in common, they are represented as separate ovals that do not touch each other, and are called disjoint sets. If two sets have one or more outcomes in common, they are represented with oval shapes that intersect each other and inside the intersection we include the common outcomes, these are called overlapping sets.

Disjoint and Overlapping Events Quiz - Teste dein Wissen

Question

What is a compound event?

A compound event of two events A and B is defined as the union of all the outcomes from both events A and B, or the intersection of the common outcomes shared by A and B.

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Question

What are disjoint or mutually exclusive events in Probability?

Disjoint or mutually exclusive events are events that have no outcomes in common, therefore they cannot occur together.

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Question

What are overlapping events in Probability?

Overlapping events are compound events with one or more outcomes in common.

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Question

What do you need to consider when working out the probability of A or B happening P(A or B)?

Consider whether the two events have any outcomes in common or not.

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Question

Is getting heads or tails when tossing a coin a disjoint or an overlapping event?

Disjoint

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Question

What type of diagrams can be used to represent disjoint and overlapping events?

Venn diagrams

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Question

How do you know if events are disjoint or overlapping?

Disjoint events have no outcomes in common, therefore they cannot occur together. Overlapping events have one or more outcomes in common.

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Question

When rolling a die, what is the probability of getting a 1 or a 3?

33.33%

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