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### Course: Digital SAT Math>Unit 3

Lesson 8: Probability and relative frequency: foundations

# Probability and relative frequency | Lesson

A guide to probability and relative frequency on the digital SAT

## What are probability and relative frequency problems?

The SAT will ask you to calculate probabilities and relative frequencies using data from two-way frequency tables.
Two-way frequency tables include two qualitative variables, one represented by rows and the other represented by columns. For example, the table below summarizes the concert attendance of students in Dr. Romelle and Dr. Yukevich's classes. The two variables are class and concert attendance.
Attended concertDid not attend concertTotal
Dr. Romelle's class$17$$14$$31$
Dr. Yukevich's class$23$$10$$33$
Total$40$$24$$64$
In this lesson, we'll learn to:
2. Use two-way frequency tables to calculate probabilities and relative frequencies
3. Use probabilities and relative frequencies to find missing values
You can learn anything. Let's do this!

## How do I read two-way frequency tables?

The biggest challenge of table data problems is understanding what the question is asking for. Let's describe some of the values in the table below.
Attended concertDid not attend concertTotal
Dr. Romelle's class$17$$14$$31$
Dr. Yukevich's class$23$$10$$33$
Total$40$$24$$64$
The number $17$ is in the row "Dr. Romelle's class" and the column "attended concert". We can make a few similar statements using the number $17$:
• $17$ students both attended the concert and are from Dr. Romelle's class.
• $17$ students from Dr. Romelle's class attended the concert.
• $17$ of the students who attended the concert are from Dr. Romelle's class.
The number $40$ is the total for the column "attended concert". This means $40$ students (from both Dr. Romelle and Dr. Yukevich's classes) attended the concert.
Similarly, the number $33$ is the total for the row "Dr. Yukevich's class". This means there are $33$ students in Dr. Yukevich's class (some attended the concert, some did not).
The number $64$ in the lower right corner is total number of students for all categories.

### Try it!

Try: identify values based on their descriptions
Owns a skateboardDoes not own a skateboardTotal
Owns a bike$4$$11$$15$
Does not own a bike$3$$7$$10$
Total$7$$18$$25$
Jackie asked his classmates whether they own a bike or a skateboard. The results are shown in the table above.
How many classmates own both a bike and a skateboard?
How many classmates own skateboards?
How many classmates own either a bike or a skateboard, but not both?

## How do I calculate probabilities and relative frequencies using two-way frequency tables?

Once we correctly identify the values we're looking for in a problem, the rest of the problem is basically dividing two values to find a fraction, percentage, or probability.
Note: While probabilities and relative frequencies are different concepts, we perform the same calculations for them.
Let's use the table below for some example calculations!
Attended concertDid not attend concertTotal
Dr. Romelle's class$17$$14$$31$
Dr. Yukevich's class$23$$10$$33$
Total$40$$24$$64$
What fraction of Dr. Romelle's class did not attend the concert?
What percent of students from both classes attended the concert?
If a student from both classes is selected at random, what is the probability that the student is from Dr. Yukevich's class and attended the concert?

### Try it!

Try: calculate a probability and a relative frequency
Owns a skateboardDoes not own a skateboardTotal
Owns a bike$4$$11$$15$
Does not own a bike$3$$7$$10$
Total$7$$18$$25$
Jackie asked his classmates whether they own a bike or a skateboard. The results are shown in the table above.
If a classmate is selected at random, what is the probability that they do not own a bike? (Enter your answer as a fraction or decimal between $0$ and $1$.)
What fraction of classmates who do not own a skateboard also do not own a bike? (Enter your answer as a fraction between $0$ and $1$.)

## How do I find missing values in a table?

Note: missing value questions appear very rarely on the SAT.
Just as we can calculate a probability or relative frequency using the values in two-way frequency tables, we can calculate missing values in a table when given a probability or relative frequency.
Some two-way frequency tables do not provide the totals for us. For these tables, it's helpful to add a row and a column for the totals.
Let's look at another example using the students in Dr. Romelle and Dr. Yukevich's classes.
Plays an instrumentDoes not play an instrumentTotal
Dr. Romelle's class$20$$11$$31$
Dr. Yukevich's class$33$
Total$64$
If $\frac{3}{4}$ of the students in the two classes play an instrument, how many students in Dr. Yukevich's class play an instrument?

### Try it!

try: find missing values in a table
B or betterC or worse
Finished book$8$$1$
Did not finish book
The table above shows the grades of $20$ students who wrote an essay based on a book. Some of the values are missing.
How many students did not finish the book?
If $50\mathrm{%}$ of the students received a B or better on the essay, how many of those students did not finish the book?

Practice: calculate a probability
Watches gaming livestreamsDoes not watch gaming livestreamsTotal
Plays video games$44$$65$$109$
Does not play video games$3$$39$$42$
Total$47$$104$$151$
Barento surveyed $151$ high school students on their video game-related activities. The results are summarized in the table above. If one student from the survey is selected at random, what is the probability that the selected student does not play video games and does not watch gaming livestreams?

Practice: calculate a relative frequency
Great BritainUnited StatesOther nationsTotal
Gold$56$$23$$31$$110$
Silver$51$$12$$44$$107$
Bronze$39$$12$$56$$107$
Total$146$$47$$131$$324$
The table above represents the medals won at the $1908$ Summer Olympics. Approximately what percent of gold medals were won by Great Britain and the United States?

Practice: calculate a missing value
Blood Type
Rhesus factor$\text{A}$$\text{B}$$\text{AB}$$\text{O}$
$+$$77$$76$$19$$122$
$-$$1$$x$$1$$3$
Human blood can be classified into four common blood types—$\text{A}$, $\text{B}$, $\text{AB}$, and $\text{O}$. It is also characterized by the presence $\left(+\right)$ or absence $\left(-\right)$ of the rhesus factor. The table above shows the distribution of blood type and rhesus factor for a group of people. If one of these people who is rhesus negative $\left(-\right)$ is chosen at random, the probability that the person has blood type $\text{O}$ is $\frac{1}{2}$. What is the value of $x$ ?

## Want to join the conversation?

• I love checking the comments after finishing every lesson 😂😂
• wallah, same😂
• whos preparing for SAT december?
• me. my current highscore is 1160 can i make it to 1400 by december??
• who is giving october SAT?
• I'm writing the october sat
• pretty easy guys
• It is
• why is the answer for the fraction question 7/18 instead of 7/25
• THe queston is: What fraction of classmates who do not own a skateboard also do not own a bike?
whenever you see question asked for a fraction, Easiest way is devide question for the fraction they are asking for.
For example they are asking for a fraction in the above question.
So We need to devide the question
Devide= Who do not own a skateboard / Do not own a bike
so Fraction is 7/18
It's a tricky way to answer these types of question.
• I feel hard to comprehend the last one "Rhesus factor".
• 3 (probability of O) is equal to half.
In probability, The total is always equal to 1.
The total here is six (3 being half)
So that means the total of the others is also 3.
2 are one each.
one left
• Who's preparing for march DSAT??
• me, redoing it
• why are box plots and leaf plots not in the lessons?
• for last question:
3/5+x=1/2

x=1