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Complete Guide to Probability on SAT Math + Practice Questions

Posted by Courtney Montgomery | Aug 28, 2017 8:30:00 AM

SAT Math



A probability question asks you to identify how likely a particular event is to occur. How likely is it that you’ll pick a red marble out of a bag? How likely is it that a particular person will be chosen out of a lottery? How likely is it that two or more events will BOTH occur? These are just some of the many different types of probability questions you may encounter on the SAT.

This guide will take you through all aspects of probability you’ll need to know for the SAT--exactly what probability means, the typical probability questions you’ll see on the SAT math section, and the steps needed to solve them.


Before You Continue

Probability questions will show up on most SAT tests. The vast majority of SAT tests only have one questions out of the 58 math questions total, although you might very occasionally see a test with 0 or 2 probability questions. So plan your SAT math study prep accordingly.

If you are struggling to understand other fundamental sections of the math test, like integers or single variable equations, you will want to turn your focus there before you tackle this probability guide. The most important part of studying for the SAT is to focus your attention on topics that show up the most. This way, you can maximize your potential point gain per section.

But if you already have a solid grasp of the other fundamental math topics (or you just really want to learn this section first), then let’s get cracking on probability! You'll learn SAT math tips and formulas to work through questions that deal with chance.


Don't worry—I hear the probability of success is higher than you'd think.


What Does Probability Mean?

$\Probability = {\desired \outcome}/{\all \possible \outcomes}$

Remember this SAT math formula! Asking for the probability of an event is the same thing as asking for the “odds” of any particular event happening. And this probability is expressed as a fraction of: the likelihood of the event over all the outcomes possible.

So how likely is it that you’ll get tails if you flip a coin? The chances are 1 in 2.

1 for the number of  outcomes you want (tails) and 2 for the total number of possibilities (heads and tails).

Let’s take a look at another example:

There are ten students in the class. Every day, the teacher selects a random student to erase the board. What are the odds that Student A will be selected to clean the board today?


The probability of Student A being selected is $1/10$. The desired outcome is 1 because Student A is only one student. And there are 10 students total, so there are 10 possible outcomes (students to pick from).


Now what would happen if we had more than one possible choice as our desired outcome?

What are the odds that either Student A or Student B will be selected to clean the board today?


The probability is now $2/10$ (or $1/5$). Why? Because there are now 2 possible students to choose from, but the total number of students is still 10.


Because the probability of any event happening is expressed as a fraction, it means that an event that will absolutely and without a doubt occur has a probability of $1/1$ or 1. There is no higher chance of it happening—this particular event will happen every single time, without fail.

A probability of an absolutely impossible event, however, will be 0 because $0/x = 0$.

You can also think of probabilities as percentages. If I select a red marble from a bag at a probability of $1/5$, it means that there is a 20% chance that I will select a red marble because $1/5 = 0.2$ or 20%.



I'm gonna go with tails on this one. 


Either/Or Probability

${\Probability \of \either \event} = [{\outcome \A}/{\total \number \of \outcomes}] + [{\outcome \B}/{\total \number \of \outcomes}]$

(Note: this kind of probability is called “non-overlapping.” This means that the two events cannot both happen at the same time. There is a way to find an either/or probability for overlapping events, but you will never be asked to do this on the SAT, so it is not in this guide)

As we saw above with our example of multiple students selected at random to clean a board, an either/or probability question asks how likely it is that either one of two or more events will occur. This increases the odds of our desired outcome because we do not care which of the two events happen, only that one of them does.

To solve this kind of problem, we must therefore add the probability of each individual event. Their sum is the probability of either event happening.

What is the probability of drawing either an ace or a queen from a deck of cards?


There are 4 aces in a deck of cards and 52 cards total. Therefore, the probability of drawing an ace is $4/52 = 1/13$ (or 7.69%).

There are also 4 queens in a deck of cards. So the probability of drawing a queen is also $1/13$.

So the probability of drawing either an ace or a queen is $1/13 + 1/13 = 2/13$ or 15.38%.


There are types of probability questions other than simple probability and either/or, but these are the only two types of probability that the SAT tests.  


Conditional Probability

Very occasionally, the SAT will hit you with a simple conditional probability question. (I found one spread across all 8 free SAT practice tests). 

Conditional probability is the chances of an event (B) happening given that another event or condition (A) has already happened or been fulfilled. It's still simple probability—desired outcomes over total outcomes—but figuring out the correct number of desired vs. total outcomes can be a little tricky. 

Here's an example:

There are 100 people working on a performance: 52 dancers, 12 stage technicians, and 36 musicians. Among the dancers, 14 are ballet dancers, 20 are jazz dancers, and 18 are modern dancers. 

What is the probability of selecting a ballet dancer from those working on the performance, given that the person selected is a dancer?


It might seem like this is asking you the probability of selecting a ballet dancer (of which there are 14) from everyone working on the performance (of which there are 100). 

But actually, it's asking you the probability of selecting a ballet dancer from the dancers, because we are accepting as a given (as a condition) that the person we are randomly selecting is a dancer. We can tell this from the phrase "given that the person selected is a dancer."  

Thus, we must calculate the probability of selecting a ballerina (Event B) given condition A, that the person we select will be from among the 52 dancers. So the answer is $14/52$. 

You can identify conditional probability questions because they will say "given" or some other word or phrase to indicate that there is some precondition being met ("provided that," "assuming," etc.).



Life would be better if there were a much higher probability of this actually happening


Typical SAT Probability Questions

Probability questions on the SAT will always be accompanied by a chart of some sort.

Here's an example from SAT practice test 1: 

Dreams Recalled During One Week: 

  None 1-4 5+ Total
Group X  15 28 57 100
Group Y   21 11 68 100
Total   36 39 125 200


The data in the table above were produced by a sleep researcher studying the number of dreams people recall when asked to record their dreams for one week. Group X consisted of 100 people who observed early bedtimes, and Group Y consisted of 100 people who observed later bedtimes. If a person is chosen at random from those who recalled at least 1 dream, what is the probability that the person belonged to Group Y?

  1. $68/100$
  2. $79/100$
  3. $79/164$
  4. $164/200$


There's no "either/or" or "given/assuming" in the question text, so we can conclude this is a simple probability question. This means we are looking for two pieces of information: the number of desired outcomes over the total number of outcomes. 

Let's actually start with our total number of outcomes: the text says we are choosing from "those who recalled at least 1 dream." So we need to figure out the total number of people (in either group) who recalled at least 1 dream. That's going to be everyone in both Group X and Group Y from the "1-4" and "5+" columns of the table.

$$28+57+11+68 = 164$$

So our total number of outcomes (or the total number of people who remembered 1+ dream) is 164.

You could also look at the "Totals" row at the bottom and add $39+125$ if that's easier for you.

Now we need to know the number of desired outcomes. The question asks us the probability that our random choice from the group of people who remembered 1+ dreams is in Group Y. So how many Group Y individuals are in our group of 164 people who remembered at least one dream?

We can figure this out by adding together the Group Y cells in the "1-4" and "5+" columns:

$$11+68 = 79$$

Our number of desired outcomes, then, is 79. 

If we put our desired outcomes (79) over our total outcomes (164) then we get $79/164$. Thus, the answer is C.



I somehow don't think the odds are that much in my favor in this game....


How to Solve a Probability Question: SAT Math Strategies

You will know if you are being asked for a probability question on the SAT because there will be a chart and the problem will ask you for the "probability of," the "proportion of," or the "odds of" one or more events happening. When you see those words, follow these two simple steps to solving a probability question:


#1: Determine What Kind of Question It Is

It's important to determine whether the question is a simple probability question, a conditional probability question, or an either/or probability question so that you know how to proceed. Remember, either/or questions will pretty much always have an "either" or an "or" in the question, and conditional probability questions will say "given" or "assuming" or some other word to indicate that there's some kind of condition or event preceding the probability you must calculate.


#2: Simplify the Idea of a Probability

Once you get used to working with probabilities, you’ll find that probability questions are often just fancy ways of working with fractions and percentages.

This is especially clear in the SAT's format, which gives you everything in a chart. You typically just need to figure out which cells of the table to add together to get the desired outcomes and put that over the cells you need to add together to get the total number of outcomes that the question is asking about. The really important part is usually identifying what the total outcomes actually are and what the desired outcomes actually are.


body_roulette.jpgWith odds of 37 to 1, how could you lose? (Answer: easily)


Test Out Your Knowledge with SAT Math Practice Questions

 1)  From SAT Practice Test 3:

  Under 40  40+ Total
Male  12 2 14
Female  3 11
 Total  20 5 25


The table above shows the distribution of age and gender for 25 people who entered a contest. If the contest winner will be selected at random, what is the probability that the winner will be either a female under age 40 or a male age 40 or older?

  1. $4/25$
  2. $10/25$
  3. $11/25$
  4. $16/25$ 


2) From SAT Practice Test 3:

  Left-handed  Right-handed
Total   18 122


The incomplete table above summarizes the number of left-handed students and right-handed students by gender for the eighth-grade students at Keisel Middle School. There are 5 times as many right-handed female students as there are left-handed female students, and there are 9 times as many right-handed male students as there are left-handed male students. If there is a total of 18 left-handed students and 122 right-handed students in the school, which of the following is closest to the probability that a right-handed student selected at random is female? (Note: Assume that none of the eight-grade students are both right-handed and left-handed.)

  1. $0.410$
  2. $0.357$
  3. $0.333$
  4. $0.250$


3) From SAT Practice Test 5:

   Teaching Research Total
General Surgeon 258  156 414
Orthopedic Surgeon  119  74 193
 Total  377 230 607


In a survey, 607 general surgeons and orthopedic surgeons indicated their major professional activity. The results are summarized in the table above. If one of the surgeons is selected at random, which of the following is closest to the probability that the selected surgeon is an orthopedic surgeon whose indicated professional activity is research?

  1. $0.122$
  2. $0.196$
  3. $0.318$
  4. $0.379$ 


4) From SAT Practice Test 7: (Grid-In Question)

Number of Contestants by Score and Day

   5/5 4/5  3/5 2/5 1/5 0/5 Total
Day 1 2 3 4 6 2 3 20
Day 2  2 3 5 5 4 1 20
Day 3 3 3 4 5 3 2 20
Total  7 9 13 16 9 6 60


The same 20 contestants, on each of 3 days, answered 5 questions in order to win a prize. Each contestant received 1 point for each correct answer. The number of contestants receiving a given score on each day is shown in the table above.

No contestant received the same score on two different days. If a contestant is selected at random, what is the probability that the selected contestant received a score of 5 on Day 2 or Day 3, given that the contestant received a score of 5 on one of the three days?


Answers: B, A, A, $5/7$


Answer Walk-Throughs

Practice Question 1: This is an either-or probability question, which we can tell because the question includes "either" and "or." So we'll be adding two simple probabilities together. 

The first probability is the chance of choosing a female under 40 at random from the group of people who entered the contest. There were 8 females under 40 and 25 entrants total according to the table, so our first probability is $8/25$. 

The second probability is the chance of choosing a male 40 or over. According to the table, there were 2 men 40 or over, and, again, we know that 25 people entered the contest, so our second probability is $2/25$.

Now we need to add our probabilities together:

$$8/25 + 2/25 = 10/25$$

Thus, the answer is B. 


Practice Question 2: Wait a second—the table isn't all the way filled out here! This is actually an algebra question and a probability question tied up in one. 

How can we figure out what goes in the table here? Well, we know that there are five times as many right-handed female students as left-handed female students. Let's say, then, that the number of left-handed female students is $f$ and the number of right-handed female students is $5f$. For the male students, we'll say that $m$ is the number of left-handed male students and then, per the problem, $9m$ is the number of right-handed male students, since there are 9 times as many. 

We also know that there is a total of 18 left-handed students and 122 right-handed students. 

This means that

$$f+m = 18$$


$$5f + 9m = 122$$

What we're really interested in here is the number of right-handed female students (which is our desired outcome), so let's solve for $f$. 

Per our first equation,


So replacing $m$ with $18-f$ in our other equation, we get:

$$5f + 9(18 - f) = 122$

$$5f + 162 - 9f = 122$$

$$-4f + 162 = 122$$

$$-4f = -40$$

$$f = 10$$


Remember, $f$ is the number of left-handed female students, and the number of right-handed female students is $5f$ or 50. This means that our desired outcome (right-handed female students) is 50.

Our total number of potential outcomes is going to be the total number of right-handed students, since that is the pool we are selecting from. That's 122. So the answer is $50/122$. In decimal form, this is 0.410. Thus, the answer is A. 



Practice Question 3: This is a good old simple probability question. We are selecting one of the surgeons at random, so the total number of potential outcomes is just the total number of surgeons—607. And we want to know the chances that a randomly-selected surgeon is an orthopedic surgeon whose professional activity is research. Per the table, there are 74 orthopedic surgeons who do research. This means our probability is $74/607$ or, as a decimal, 0.122. So the answer is A.  


Practice Question 4: This grid-in question is both an either/or probability question and a conditional probability question. The condition is we are selecting at random among contestants who received a 5/5 on one of the three days. The question tells us that we can assume that no one got the same score multiple days, so we can assume that the 7 people who received 5s across the 3 days are all different people. 

Our total number of potential outcomes, then, is 7—not 20, the total number of contestants. 

Our desired outcome is the number of people receiving 5s on Day 2 or Day 3 (there's that either/or)! 2 people received 5s on Day 2, and 3 on Day 3. Our two probabilities that we are adding together, then, are

$$2/7 + 3/7 = 5/7$$

The answer, then, is $5/7$.



Whoo! Take your new probability knowledge and celebrate!


The Take-Aways

Probability questions can seem much trickier than they actually are. By taking the time to analyze what is being asked of you—what's really the desired outcome? How many possible outcomes are there really? Is this an either/or question?—and understanding that probabilities are simply fractional relationships of desired outcomes over all potential outcomes, you’ll be able to tackle SAT probability questions in no time.


What’s Next?

You’ve stacked the odds in your favor by mastering SAT probability. Now that you’re done, it’s probably a good idea to take a look at all the topics covered on SAT math.

Don’t know how you could possibly finish a math section on time? Look no further than our article on how to buy yourself time and complete your SAT math problems before it’s pencils down.

Want to get a perfect score? If you’ve already mastered your timing and score, it may be time to look at our article on how to get a perfect 800 on SAT Math, written by a perfect-scorer.


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Courtney Montgomery
About the Author

Courtney scored in the 99th percentile on the SAT in high school and went on to graduate from Stanford University with a degree in Cultural and Social Anthropology. She is passionate about bringing education and the tools to succeed to students from all backgrounds and walks of life, as she believes open education is one of the great societal equalizers. She has years of tutoring experience and writes creative works in her free time.

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