AP®StatisticsCollege BoardStudy GuidesUnit 4: Probability, Random Variables & Probability DistributionsProbabilityProbabilities of Single Events

Probabilities of Single Events (College Board AP® Statistics)

Study Guide

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Dan Finlay

Written by: Dan Finlay

Reviewed by: Lucy Kirkham

Sample space

What is a sample space?

  • A sample space is the set of all possible outcomes of a random process

    • The outcomes do not overlap

  • A sample space is written using set notation

    • e.g. the sample space for rolling a six-sided dice is {1, 2, 3, 4, 5, 6}

  • Two-way tables and tree diagrams can help to find all outcomes for a sample space when multiple factors are being considered

    • e.g. a two-way table can be used to find the sample space when a dice is rolled and a coin is flipped

Table showing sample space of flipping a coin (H for heads, T for tails) and rolling a dice (numbers 1 to 6). Resulting pairs: H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6.
Sample space for rolling a dice and flipping a coin

Calculating probabilities of events

How do I find the probability of an event when the outcomes are equally likely?

  • If the outcomes are equally likely then you can find the probability of an event A using the formula

    • P open parentheses A close parentheses equals fraction numerator number space of space outcomes space in space event space A over denominator total space number space of space outcomes space in space sample space space end fraction

Worked Example

Two fair coins with faces labeled heads (H) and tails (T) are independently flipped. Find the probability that the two coins land on faces with different labels.

Answer:

Find the sample space

The sample space is open curly brackets H H comma space H T comma space T H comma space T T close curly brackets

Identify the outcomes that are in the event

Let A represent the event that both coins land on faces with different labels

A equals open curly brackets H T comma space T H close curly brackets

The outcomes are equally likely because the coins are fair and flipped independently
Find the probability by using the formula P open parentheses A close parentheses equals fraction numerator number space of space outcomes space in space event space A over denominator total space number space of space outcomes space in space sample space space end fraction

P open parentheses A close parentheses equals 2 over 4

The probability that the two coins land on faces with different labels is 1 half

How do I find the probability of an event when the outcomes are not equally likely?

  • If the outcomes are not equally likely then you can find the probability of an event A by adding together all the probabilities of the outcomes in event A

Worked Example

A biased four-sided dice has faces labeled 1 to 4. The dice is rolled and the face it lands on is recorded. The dice landing on a 1, 2, 3 or 4 has a probability of 0.1, 0.2, 0.3 or 0.4 respectively.

Find the probability that the dice lands on a face labeled with a number less than 4.

Answer:

Identify the outcomes that are in the event

Let A represent the event that the dice lands on a face labeled with a number less than 4

A equals open curly brackets 1 comma space 2 comma space 3 close curly brackets

The outcomes are not equally likely so add the probabilities of each outcome in the event

table row cell P open parentheses A close parentheses end cell equals cell P open parentheses 1 close parentheses plus P open parentheses 2 close parentheses plus P open parentheses 3 close parentheses end cell row blank equals cell 0.1 plus 0.2 plus 0.3 end cell row blank equals cell 0.6 end cell end table

The probability that the dice lands on a face labeled with a number less than 4 is 0.6

Complement of an event

What is the complement of an event?

  • The complement of an event A is the set of outcomes that are not in the event A

    • If event A is a dice landing on an even number when rolled

      • then its complement is the event that the dice lands on an odd number

  • The complement of an event A is denoted by A apostrophe or A to the power of C

How do I find the probability of the complement of an event?

  • The probabilities of an event and its complement add up to 1

  • Subtract the probability of an event from 1 to find the probability of its complement

    • P open parentheses A apostrophe close parentheses equals 1 minus P open parentheses A close parentheses

Examiner Tips and Tricks

If you see the phrase "at least" in an exam question, then consider using the complement of an event.

Worked Example

Jim plays a game once each day. The probability that Jim first wins the game on day 1 is 0.5. The probability that Jim first wins the game on any given day is half the probability that Jim first wins the game on the previous day. For example, the probability that Jim wins the game on day 2 is 0.25.

Find the probability that Jim does not win his first game until at least day 3.

Answer:

Identify the event

Let A represent the event that Jim does not win his first game until at least day 3

The event has an infinite number of outcomes as Jim could win his first game on day 3, day 4, day 5, and so on

P open parentheses A close parentheses equals P open parentheses first space wins space on space day space 3 close parentheses plus P open parentheses first space wins space on space day space 4 close parentheses plus P open parentheses first space wins space on space day space 5 close parentheses plus...

The missing outcomes are that Jim first wins on day 1 and that Jim first wins on day 2, this is the complement of the event in the question

A apostrophe is the event that Jim first wins the game on day 1 or day 2

Find the probability of the complement by adding together the probability that Jim first wins on day 1 and the probability that Jim first wins on day 2

table row cell P open parentheses A apostrophe close parentheses end cell equals cell P open parentheses first space wins space on space day space 1 close parentheses plus P open parentheses first space wins space on space day space 2 close parentheses end cell row blank equals cell 0.5 plus 0.25 end cell row blank equals cell 0.75 end cell end table

Subtract this probability from 1 to find the probability that Jim does not win his first game until at least day 3

table row cell P open parentheses A close parentheses end cell equals cell 1 minus P open parentheses A apostrophe close parentheses end cell row blank equals cell 1 minus 0.75 end cell row blank equals cell 0.25 end cell end table

The probability that Jim does not win his first game until at least day 3 is 0.25

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    Author
    Reviewer
    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

    Lucy Kirkham

    Author: Lucy Kirkham

    Expertise: Head of STEM

    Lucy has been a passionate Maths teacher for over 12 years, teaching maths across the UK and abroad helping to engage, interest and develop confidence in the subject at all levels.Working as a Head of Department and then Director of Maths, Lucy has advised schools and academy trusts in both Scotland and the East Midlands, where her role was to support and coach teachers to improve Maths teaching for all.