AP®StatisticsCollege BoardRevision NotesUnit 4: Probability, Random Variables & Probability DistributionsDiscrete Random VariablesProbability Distributions for Discrete Random Variables

Probability Distributions for Discrete Random Variables (College Board AP® Statistics)

Revision Note

Dan Finlay

Expertise

Maths Lead

Discrete probability distributions

What is a random variable?

  • A random variable is a variable whose values represent the outcomes of a random event

    • e.g. a random variable could represent the height of a person or the number of hats owned by a person

  • The word random does not refer to the probability of each outcome occurring

    • It means the variable can take any value from its possible outcomes

    • The outcome is not pre-determined

  • Random variables are denoted by uppercase letters (X, Y, etc)

  • The outcomes are denoted by lowercase letters (x, y, etc)

  • The probability that the random variable X takes the value x is denoted as P open parentheses X equals x close parentheses or P open parentheses x close parentheses

What is a discrete random variable?

  • A discrete random variable is a random variable with a countable number of values

    • The number of values could be finite

      • e.g. the number of times a coin lands on heads when flipped 10 times,
        the outcomes are {1, 2, 3, ..., 10}

    • The number of values could be infinite

      • e.g. the number of times a coin is flipped until it first lands on heads,
        the outcomes are {1, 2, 3, ...}

What is a discrete probability distribution?

  • A discrete probability distribution describes the probabilities of the occurrences of each possible outcome associated with a discrete random variable

  • A discrete probability distribution can be given in several forms:

    • a table

    • a function

    • or a graph

A diagram showing a scenario of an eight-sided spinner with numbers 0, 1/3, 5, -2, and the probability distribution of the scenario in three forms. A table showing probabilities of the spinner landing on each number, P(0)=1/8, P(1/3)=1/8, P(5)=1/2, P(-2)=1/4; a function describing the same probabilities and a bar graph of these values.
Example of the ways to display a probability distribution for a discrete random variable

  • You can interpret a probability distribution in its graphical form by looking at:

    • its shape

    • its center

    • and its spread

  • If the outcomes are equally likely then the bars on the graph have the same heights

    • These are called discrete uniform distributions

How can I use a discrete probability distribution?

  • The sum of all the probabilities is equal to 1

    • You can use this to find a missing probability

  • To find the probability of an event

    • Identify the outcomes which are in the event

    • Add together the probabilities of those outcomes

Exam Tip

If you introduce a random variable when answering an exam question, make sure you state what the variable represents.

For example, let the random variable X represent the number of points scored in an exam.

Worked Example

Each day, Josh drives through four sets of traffic lights on his way to work. The random variable, X, represents the number of sets of traffic lights that Josh has to stop at on his way to work on a randomly selected day. The table below shows the probability distribution of X.

x

0

1

2

3

4

P open parentheses X equals x close parentheses

0.4

0.3

P open parentheses X equals 2 close parentheses

0.1

0.05

(a) Calculate the probability that Josh has to stop for exactly 2 sets of traffic lights on his way to work on a randomly selected day.

Answer:

The required probability is P open parentheses X equals 2 close parentheses

The probabilities of the outcomes add up to 1

Form an equation and solve it

table row cell 0.4 plus 0.3 plus P open parentheses X equals 2 close parentheses plus 0.1 plus 0.05 end cell equals 1 row cell P open parentheses X equals 2 close parentheses plus 0.85 end cell equals 1 row cell P open parentheses X equals 2 close parentheses end cell equals cell 0.15 end cell end table

The probability that Josh has to stop for exactly 2 sets of traffic lights on his way to work on a randomly selected day is 0.15

(b) Calculate the probability that Josh has to stop for at least one set of traffic lights on his way to work on a randomly selected day.

Answer:

The required probability is P open parentheses X greater or equal than 1 close parentheses

At least once means Josh could stop at 1, 2, 3 or 4 traffic lights

This is all of the outcomes except 0

Either add the probabilities for X equals 1 comma 2 comma 3 comma 4 or subtract the probability for X equals 0 from 1

table row cell P open parentheses X greater or equal than 1 close parentheses end cell equals cell P open parentheses X equals 1 close parentheses plus P open parentheses X equals 2 close parentheses plus P open parentheses X equals 3 close parentheses plus P open parentheses X equals 4 close parentheses end cell row blank equals cell 0.3 plus 0.15 plus 0.1 plus 0.05 end cell row blank equals cell 0.6 end cell end table

or

table row cell P open parentheses X greater or equal than 1 close parentheses end cell equals cell 1 minus P open parentheses X equals 0 close parentheses end cell row blank equals cell 1 minus 0.4 end cell row blank equals cell 0.6 end cell end table

The probability that Josh has to stop for at least one set of traffic lights on his way to work on a randomly selected day is 0.6

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    Dan Finlay

    Author: Dan Finlay

    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.