ACT Math: Guide to Stats & Probability
tl;dr: If you are looking for help in understanding statistics and probability questions on the ACT Math Section, this article has you covered. We will go over several strategies and tips, such as basic probability, combinations, statistical measures, and expected value. Remember to keep replacement in mind and multiply possibilities for combinations. Also, know that the mean, median, mode, and range are key statistical measures. Finally, understand expected value and how it is calculated to help you make the best decisions on the ACT.
What are the odds? Is this a fair game? What are the potential winnings? These are all questions that we often ask throughout our everyday life. Also, these same mathematical concepts appear on the ACT in the math section, so you are expected to know how to apply many of these concepts to make a good analysis and statistical decisions. This article will be going over several ACT strategies, tips, and tricks to help you make it through the statistics questions.
The good news is that ACT very clearly publishes the breakdowns of its exam per subject. For statistics and probability, you can expect to see 5-7 questions, totaling 8-12% of the total math portion of the exam. In this guide, we will look at some of the central ideas you can expect to see on your ACT exam. If you're new to how the ACT works, be sure to check out this article that summarizes what you will find on this test.
Basic Probability ๐ฒ
The first type of problem you will likely encounter regarding statistics and probability will involve basic probability. These are probably very familiar problems to you and will more than likely appear in the first 15 questions of the test.
The most important thing to remember for this type of question is how many total possibilities there are and how many ways you can successfully draw what is tested. For instance, if we have a jar of marbles and we are looking at the probability of drawing a blue ๐ต marble, take the total marbles as your denominator and the total number of blue marbles as the numerator. Sometimes probabilities as fractions are simplified, sometimes they are not. A handy scientific calculator is a great way to make this process simple and fast. ๐
Things to Remember
When considering calculating basic probability questions, there are a few things you need to be aware of:
- Remember to consider replacement if there are multiple events. If we are drawing multiple marbles and not replacing them, the denominator changes as we progress since there is a lower total.
- If we are calculating the probability of multiple events occurring at the same time, we multiply the probability of each event. Again, keep replacement in mind as you are calculating!
- Keep in mind certain conditions. Sometimes you may be asked to find the probability of finding a certain probability GIVEN that another condition has already occurred, which lowers the denominator for your probability.
Example
A class is in charge of selecting a representative for the prom committee. The class consists of 87 students, including a president, vice president, and secretary. Assuming that one of the officers CANNOT be chosen for the committee, what are the odds that Dustin is randomly chosen to represent his class?
This question features conditional probability, which is key to the solution. The situation tells us that the three officers CANNOT ๐ ๐พ be the representative and, therefore, the total group we are pulling from is not 87 students, but 84. So the odds that Dustin is chosen at random is 1/84.
Combinations ๐ฏโโ๏ธ๐ฏ๐ฏโโ๏ธ
Combinatorics is also a common type of problem you might encounter on the ACT within the first 15 questions or so, and it can get you some easy points!
You will likely see problems regarding putting together a combination of multiple items to make a unique selection. A lot of times, this involves food menus or putting together committees (similar to above).
Things to Remember
The biggest thing to remember with combinations is that, usually, you should multiply the total number of possibilities within each variable to figure out how many unique selections there are.
Example
At the Hawkins Ice Cream Social ๐ฆ there are five flavors to choose from, two types of cones, and four additional toppings. Assuming you are limited to choosing one type of cone, one flavor ice cream, and one topping, how many unique ice cream cones can one create?
These questions are very common and very easy. We all love easy points! 5x2x4=40. That means there are 40 ice cream combinations! That's it! This also saves you time and gives you valuable time to work on other problems!
Statistical Measures: Mean, Median, Mode, and Range ๐ถ
Another important statistical concept that you can expect to see on the ACT is statistical measures such as mean, median, mode, and range. Statisticians can tell a lot about a set of data using these measures, which is why the ACT commonly tests on these.
What are they?
Once you figure out what each of these measures represent and how to find them, you will be well on your way to acing this section of the ACT!
- Range: This measures the spread of the data. It is found by taking the maximum value and subtracting the minimum.
- Mean: This is otherwise known as the average. Simply add up all your data points and divide by the number of data points!
- Median: This is the number in the middle. First, you must order your data from least to greatest. If there is an odd number of data points, you can find the number precisely in the middle. If there is an even number, take the average of the middle 2 data points.
- Mode: This is the number that appears the most. It also shows the peak in a distribution graph like a frequency chart or a density curve.
Example
Eliza scored an 80, 82, 100, and 95 on her first four tests. What must Eliza make on the last test to have a mean test score of 90?
There are several ways to work this one, and one of the most common methods is to work backward by using the answer choices until you get that test average of 90.
Another way to find the answer is to think about the sum of the 5 test scores if it reached an average of exactly 90. All 5 test scores would be a 90 for the average to equal 90, so they would sum up to 450. Right now, our 4 test scores sum to 357. T
Therefore, we can subtract the current sum of 357 from our needed 450 to get 93 as the minimum test score needed for Eliza to get an A.
Expected Value ๐คท๐ฟโโ๏ธ
Another trending topic on the ACT in recent years is finding the expected value of a given event. In reality, this is another way of asking to find the mean.
Expected value most often shows up when there is a game, and you are trying to find the expected earnings when you play the game for a certain amount of time. (Spoiler alert: Slot machines have a small and negative expected value, so, in the long run, the player loses. Sorry y'all ๐) A game with an expected value of 0 is considered a fair game since both sides have an equal chance of gaining or losing.
If presenting a statistical game, the problem will likely describe the game or give a table of possible outcomes, along with the probabilities of each.
How Do I Find It?
To find the expected value of a given game or event, you will take each outcome and multiply it by the probability of that given event (either from the table or the game description). Then you will add up each of those products for every possible outcome to get the expected value.
Some common games only have two outcomes (win and lose); others can have multiple outcomes if it is a discrete variable with the possibility of winning 1, 2, 3, etc., items.
Example
Also, note that the true outcomes are winning $2 and $7 since she has to pay $3 to play.
Now to find the expected value: -3(0.6)+2(0.35)+7(0.05)= -$0.75. This means that Eliza will lose $0.75 on average each time she plays. Therefore, this is not a fair game, and Eliza should probably not play. If she were to play 100 times, she would lose approximately $75. Carnival odds...ยฏ\_(ใ)_/ยฏ
Closing
Give yourself a pat on the back for making it this far ๐ We went through some tips on geometry questions, as well as thoroughly walked through some example questions ๐ถโโ๏ธ. If you need extra ACT help, check out this list of awesome resources, or the ultimate ACT study guides listed below to help you strive for that 36!
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Need more ACT practice?
Fiveable has you covered! Check out these articles that tell you all you need to know about each ACT Subject!
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