Back to Exercise: Define a random variable and graph its distribution

Exercises: Define a Random Variable and Graph Its Probability Distribution

Work through each section in order. A random variable X is a rule that
assigns a NUMBER to each outcome in a sample space. To build its probability
distribution, group the outcomes by the value of X and add the probabilities
in each group; the probabilities of all values must sum to 1. Write
probabilities as fractions or decimals. For explanation problems, use
complete sentences. Do NOT compute expected value — that comes next lesson.

Grade 11·19 problems·~35 min·Common Core Math - HS Statistics and Probability·group·hss-md-a-1

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Warm-Up: Sample Spaces and Probabilities

These problems review sample-space ideas you already know.

Two fair coins are tossed, with sample space $\{$ HH, HT, TH, TT $\}$ . Let $X = $ the number of heads. Which statement correctly describes what $X$ does?

$X$ assigns a NUMBER to each outcome, e.g. $X(\text{HH}) = 2$ and $X(\text{TT}) = 0$ .

$X$ equals the outcome itself, e.g. $X = \text{HH}$ .

$X$ is the probability of getting heads, $\frac{1}{2}$ .

$X$ is the list of outcomes $\{$ HH, HT, TH, TT $\}$ .

A fair die is rolled and the sample space is $\{1, 2, 3, 4, 5, 6\}$ , each outcome equally likely. What must be true of the probabilities of all six outcomes?

They are each $\frac{1}{6}$ and they sum to $1$ .

They are each $\frac{1}{6}$ and they sum to $6$ .

They are each $1$ , since every face can appear.

They cannot be found without rolling the die many times first.

Two fair coins are tossed; the sample space is $\{$ HH, HT, TH, TT $\}$ with each outcome equally likely. Let $X = $ the number of heads. Give $X$ for each outcome: $X(\text{HH}) = $ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ , $X(\text{HT}) = $ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ , $X(\text{TT}) = $ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .

Fluency Practice

Build distributions by grouping outcomes by value and summing probabilities.

Two fair coins are tossed; each of the four outcomes $\{$ HH, HT, TH, TT $\}$ has probability $\frac{1}{4}$ . Let $X = $ the number of heads. List the possible VALUES of $X$ from least to greatest: ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ , ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ , ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .

Two fair coins are tossed; each outcome $\{$ HH, HT, TH, TT $\}$ has probability $\frac{1}{4}$ . Let $X = $ the number of heads. The value $X = 0$ comes only from the outcome TT. What is $P(X = 0)$ ? Enter your answer as a fraction.

Two fair coins are tossed; each outcome $\{$ HH, HT, TH, TT $\}$ has probability $\frac{1}{4}$ . Let $X = $ the number of heads. The value $X = 1$ comes from BOTH HT and TH. What is $P(X = 1)$ ? Enter your answer as a fraction.

A family has three children; the eight equally likely outcomes give the number of girls $X$ this distribution:

Value $x$	0	1	2	3
$P(X = x)$	$\frac{1}{8}$	$\frac{3}{8}$	$\frac{3}{8}$	$\frac{1}{8}$

Which check confirms this is a valid probability distribution?

The probabilities sum to $1$ : $\frac{1}{8} + \frac{3}{8} + \frac{3}{8} + \frac{1}{8} = \frac{8}{8} = 1$ .

The values $0, 1, 2, 3$ sum to $6$ .

The largest probability is $\frac{3}{8}$ , which is less than $1$ .

There are four different values of $X$ .

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