Unit 2 · Topic 2.8 Beta

Introduction to Random Variables and Probability Distributions

A random variable is a number from a chance process, defined in context.

Practice 3: Analyze Data

Part 1 · Hook

Why this matters

A highway agency wants to know how many people ride in each car, to plan a carpool lane. One car is chosen at random: it might hold 1 person, or 5. That uncertain count is a random variable, and listing its values with their probabilities answers every question the agency has.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. A die is rolled. What is P(roll is at least 5)?

$\frac{2}{6}$
$\frac{1}{6}$
$\frac{5}{6}$
$\frac{4}{6}$

Show the answer

5 and 6: 2 of 6 outcomes.

Correct: $\frac{2}{6}$ :
$\frac{1}{6}$ :
$\frac{5}{6}$ :
$\frac{4}{6}$ :

2. Which variable is discrete?

The number of siblings a student has
A student's height
The time to finish a test
The weight of a backpack

Show the answer

Siblings are counted, with gaps between the values.

Correct: The number of siblings a student has:
A student's height:
The time to finish a test:
The weight of a backpack:

Part 4 · See it

See it first

Probability histogram of X, the number of people in a car at a toll booth: P(X = 1) = 0.45, P(X = 2) = 0.30, P(X = 3) = 0.12, P(X = 4) = 0.09, P(X = 5) = 0.04. The bars add to 1. — The probability distribution of X, the number of people in a car. The bars add to 1. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

A chance process produces a numberthat number is a random variable, named with a capital letter and defined in context
List every value with its probabilitythe probability distribution, a table or a probability histogram
Every outcome gives exactly one valuethe probabilities are from 0 to 1 and add to 1, which finds a missing value
Add the probabilities up to a boundarya cumulative probability such as $P(X \le 3)$ ; translate "fewer than 3" as $X \le 2$

Part 6 · Key ideas

Key ideas

A random variable is a number that comes from a chance process. Define it in context: "Let X = the number of people in a randomly chosen car."
A probability distribution gives every value of the random variable and its probability, as a table or a probability histogram. The probabilities are from 0 to 1 and add to 1.
A discrete random variable has separate values (counts); a continuous one can take any value in an interval.
A cumulative probability adds up to a boundary: $P(X \le 3)$ . "Fewer than 3" is $X \le 2$ ; "at least 3" is $X \ge 3$ .

Part 7 · Misconception

A common mistake

The wrong idea: "Let X = rock songs" defines the random variable.

What actually happens: A random variable is a number. Say what is counted and out of what: "Let X = the number of rock songs among the 12 songs on the playlist."

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Table

People in a car at a toll booth

A highway agency records the number of people in each car that passes a toll booth. Let X = the number of people in a randomly chosen car. The probability for 5 people (the most a car can hold here) is missing (invented data).

Probability distribution of X, the number of people in a car
$x$	1	2	3	4	5
$P(X = x)$	0.45	0.30	0.12	0.09	?

1. What probability belongs in the blank, $P(X = 5)$ ?

Type a number. A table answer and a calculator answer are both accepted.

Show the answer

The probabilities of a distribution add to 1: $1 - (0.45 + 0.30 + 0.12 + 0.09)$ $= 1 - 0.96$ $= 0.04$ .

Answer: 0.04

2. What is the probability that a randomly chosen car has at least 3 people?

Type a number. A table answer and a calculator answer are both accepted.

Show the answer

"At least 3" is 3, 4 or 5: $P(X \ge 3)$ $= 0.12$ $+ 0.09$ $+ 0.04$ $= 0.25$ .

Answer: 0.25

3. What is the probability that a car has fewer than 3 people?

Type a number. A table answer and a calculator answer are both accepted.

Show the answer

"Fewer than 3" means 1 or 2, that is $X \le 2$ : $P(X \le 2)$ $= 0.45 + 0.30$ $= 0.75$ .

Answer: 0.75

4. Which is a complete definition of the random variable in this context?

X = the people in the cars at the toll booth
X = the number of people in a randomly chosen car at the toll booth
X = the probability that a car has more than one person
X = the cars that pass through the toll booth on a randomly chosen day

Show the answer

A random variable is a number that comes from a chance process, defined in context: what is counted, and for which randomly chosen individual.

X = the people in the cars at the toll booth: People are not a number. Say what is counted: the number of people.
Correct: X = the number of people in a randomly chosen car at the toll booth: Right: a count, in context, for one random car.
X = the probability that a car has more than one person: X takes values like 1 to 5; probabilities describe how likely each value is.
X = the cars that pass through the toll booth on a randomly chosen day: Cars are the individuals, not the numerical value being recorded.

Graph

Goals scored by a soccer team

Let G = the number of goals a club soccer team scores in a randomly chosen game. The probability histogram shows its distribution, built from several seasons of results (invented data).

5. What is the probability that the team scores at least 2 goals in a game?

Type a number. A table answer and a calculator answer are both accepted.

Show the answer

$P(G \ge 2)$ $= 0.22$ $+ 0.12$ $+ 0.06$ $= 0.40$ , or $1 - P(G \le 1)$ $= 1 - 0.60$ $= 0.40$ .

Answer: 0.4

6. What kind of random variable is G?

Continuous, because a game lasts for 90 minutes of playing time
Discrete, because it takes separate whole-number values
Categorical, because each bar is a category
Continuous, because it can be any number from 0 to 4

Show the answer

G takes a list of separate values (0, 1, 2, 3, 4), so it is a discrete random variable.

Continuous, because a game lasts for 90 minutes of playing time: The variable is goals, not time.
Correct: Discrete, because it takes separate whole-number values: Right: goals are counted.
Categorical, because each bar is a category: G is a number from a chance process; the bars are its values.
Continuous, because it can be any number from 0 to 4: It cannot be 2.5 goals; only whole numbers are possible.

Part 9 · Summary

Summary

A random variable is a number from a chance process, defined in context. Its probability distribution lists each value with its probability; the probabilities are from 0 to 1 and add to 1. Probabilities of events like "at least 3" or "fewer than 3" come from adding the right values, and cumulative probabilities add everything up to a boundary.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic