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Sample spaces and events

Describe a random experiment by its sample space, the set of all possible outcomes, and treat events as subsets of it. Then write compound events as set expressions: exactly one of two events, none of several events, and not all of them.

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Exactly one of two events

Learn how to write the event that exactly one of two events occurs as a set expression, either as one event occurring without the other or as the union with the overlap removed.

(A \cap B^c) \cup (A^c \cap B)

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Lesson 03 · Exactly one of two events

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A

B

Step 1: $A$ occurs, $B$ does not

The shaded region holds the outcomes in $A$ only: in $A$ but not in $B$ . This is the event $A \cap B^{c}$ .

It is also the set difference $A ∖ B$ , read " $A$ minus $B$ ".

Step 2: $B$ occurs, $A$ does not

The shaded region holds the outcomes in $B$ only: in $B$ but not in $A$ . This is the event $A^{c} \cap B$ .

As a set difference, it is $B ∖ A$ .

Step 3: Exactly one occurs

Joining the two regions gives the event that exactly one of $A$ , $B$ occurs:

(A \cap B^{c}) \cup (A^{c} \cap B)

In set-difference notation, this is $(A ∖ B) \cup (B ∖ A)$ . The overlap, where both events occur, is left out.

Step 4: Compare with $A \cup B$

The union $A \cup B$ contains both regions and also the overlap $A \cap B$ , where both events occur.

So $A \cup B$ is larger than the event that exactly one occurs.

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The 4 lessons

The 4 lessons in this unit. Expand any one to see what it covers. Each unlocks when its own prerequisites are passed.

Random experiments and outcomes

12 questions

Learn how to identify random experiments versus deterministic ones, define outcomes and sample spaces using set notation, and construct sample spaces for multi-stage random experiments.

Covers

What is a random experiment

Outcomes and sample spaces

Multi stage experiments

Events as sets of outcomes

8 questions

Learn how to represent events as subsets of a sample space, distinguish between simple and compound events, and identify the certain event and the impossible event as the two extremes of any probability model.

Covers

Events as subsets

The certain and impossible events

Exactly one of two events

Preview

8 questions

Learn how to write the event that exactly one of two events occurs as a set expression, either as one event occurring without the other or as the union with the overlap removed.

Covers

Exactly one of $A$ , $B$

Exactly one as union minus overlap

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None versus not all

9 questions

Learn how to write the event that none of several events occurs, and the event that not all of them occur, as set expressions built from complements, and tell the two apart.

Covers

None of the events

Not all of the events

Where this unit sits in the curriculum

This unit is part of our Essential Probability & Statistics for ML learning path, which contains 15 lessons. Every one of them is drawn below.

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4 lessons in this unit

11 prerequisite lessons across the unit, counting every step back to the start

Every dot is a lesson, every line a prerequisite. You can only start a lesson once you have mastered all of its prerequisites, so you are always building on solid foundations.

Builds on: Set operations, Summation, product, and indexed notation

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See it taught

Exactly one of two events

Step 1: A occurs, B does not

Step 2: B occurs, A does not

Step 3: Exactly one occurs

Step 4: Compare with A∪B

The 4 lessons

Where this unit sits in the curriculum

Start Sample spaces and events

Step 1: $A$ occurs, $B$ does not

Step 2: $B$ occurs, $A$ does not

Step 4: Compare with $A \cup B$