[fs-toc-h2]Probability
Probability is simply the chance of something occurring. On the SAT, this is usually defined as a fraction where the numerator (top number) is the number of chances a single event can occur and the denominator is the total number of events that can occur.
For example, let's say we have a bag of Sour Patch Kids. In this bag, there are 7 red, 3 green, 8 yellow, and 2 blue sour patch kids, making a total of 20. The probability of choosing a yellow sour patch kid would be the number of chances we could select a yellow sour patch kid (8) over the total amount ofsour patch kids (20)
\[
\frac{8}{20} = \frac{2}{5} \text{ (simplified)}.
\]
If I wanted to know the chances of pulling out a yellow or blue sour patch kid, I would add up the total number of chances of getting yellow or blue, and again put it over the total possible outcomes (20):
\[
\frac{8 + 2}{20} = \frac{10}{20} = \frac{1}{2}.
\]
Probabilities may also be defined as a decimal or percent; for example,
\[
\frac{1}{2} \text{ can be written as } 0.5 \text{ or } 50\%.
\]
(Most of the time, it will be in fraction or decimal form.)
This quick video expands further on the basics of probability.
SAT probability problems are not mathematically challenging. From a pure math standpoint, they’re some of the easiest problems. What makes these questions challenging is the language used and figuring out what the denominator and numerator are.
Probability questions are usually long paragraphs with a lot of reading, and if you misread a single word, there's a high chance you'll get the question wrong. With these questions, after you've learned the basics of probability, the most important skill is reading carefully.
[fs-toc-h4]Example 1
The first step to every probability question is determining the denominator/the total sample size. This is almost always determined by the first sentence in the question: the ‘if’ statement. The if statement in this question is: “If one of these students is selected at random”. Understanding what this sentence is really saying is incredibly important.
Notice how it tells us “if one of these students is selected at random”, not “one of the eighth grade students” or “one of the sixth grade students”. This is basically telling us to take into account every single student, which is the value listed under the 'total' in the table and/or directly given at the beginning of the question. In this case, it's 80 students, so 80 will be our denominator.
We’re looking for “the probability of selecting a student whose vote for new mascot was for a lion”, meaning our numerator will be the number of students who voted for the lion.
The intersection of the lion row and the total column will be the total number of students who voted for the lion. In this case, it’s 20.
Our denominator is 80 and our numerator is 20 giving us the fraction:
\[
\frac{20}{80} = \frac{1}{4}
\]
making answer choice C the correct answer.
[fs-toc-h4]Example 2
Again, the first thing we’ll do is determine the denominator. Our if statement is “If one of the surgeons is selected at random…” It's not specifically asking about orthopedic or general surgeons, but any surgeon. This means our total/denominator will be the total amount of all surgeons, which is 607.
Now, onto the numerator. The question asks which of the following is the closest to the probability that the selected surgeon is an orthopedic surgeon whose indicated professional activity is research. So, our numerator is going to be the number of orthopedic surgeons who primarily do research. Line up the orthopedic row with the research column to determine that there are 74 orthopedic surgeons with their major activity being research. This makes our numerator 74.
Our final fraction ends up being:
\[
\frac{74}{607}
\]
which, when divided out, gives us a probability of approximately 0.122, making answer choice A the correct answer.
[fs-toc-h4]Example 3
This question is a bit more challenging than the others, as you almost have to work backwards. They give you the probability and ask you to find a data point.
Don’t get scared or freeze up though, all the concepts are the exact same, simply tested in a different way.
Let's determine our denominator by looking for the ‘if’ statement: “If one of these people who is rhesus negative (-) is chosen at random…”. Our denominator is then the total number of people who have a rhesus negative blood type, which is the second row, so adding all these numbers up we will get our total number and denominator. The 4 columns (A, B, AB, O) have the respective values of 7, 2, 1, and \( x \). The sum of these numbers is \( 7 + 2 + 1 + x \) or \( 10 + x \), meaning our denominator is \( 10 + x \).
Now for the numerator. Instead of asking us the probability of someone having blood type B, they give us the probability and ask us to find x from that. So what we can do is compare the given probability of choosing a B- to our ‘unfinished’ probability using \( 10 + x \) as the denominator in order to solve for \( x \). Since 2 people are B-, the probability of someone being B- is:
\[
\frac{2}{10 + x}.
\]
We’re also given the full numerical value of the actual probability which is \( \frac{1}{9} \). So we can say that:
\[
\frac{1}{9} = \frac{2}{10 + x}.
\]
Use cross multiplication to solve for \( x \):
\[
10 + x = 18
\]
\[ x = 8\]
Making 8 the correct answer.
Videos:
https://youtube.com/watch?v=9Mu9Miy-Kc8
https://www.youtube.com/watch?v=7UlnSeTW0N4
note: a question like q. 100 will not show up on the SAT)