Math Study Guide for the SAT Exam
Page 10
Probability and Relative Frequency
The concepts of probability and relative frequency are closely related, so they are discussed here in one section.
Probability
Probability is the likelihood of an event occurring. The value is given as a fraction or a decimal number, and it is always less than \(1\). The closer the value is to \(1\), the higher its probability of occurring. Another way of describing this concept is to say that the probability of an event occurring is the ratio of desired outcomes (\(n(E)\)) to the total number of possible outcomes (\(n(\varepsilon\)). As a formula, we can convey this as:
\[P(E) = \frac{n(E)}{n(\varepsilon)}\]For instance, to determine the probability of randomly picking a green shirt from a hamper that contains two green shirts, three red shirts, five yellow shirts, and five blue shirts, we apply the formula and calculate the probability:
\[P (G) = \frac{2}{15}\]There is a \(\frac{2}{15}\) or \(0.133\) chance of picking a green shirt.
To determine the probability of two or more events, we multiply their individual probabilities. In the same example, we calculate the probability of randomly picking a green shirt and then a red shirt as follows:
\[P (G+R) = \frac{2}{15} \cdot \frac{3}{14} = \frac{6}{210} = \frac{1}{35}\]Note that when determining the probability of choosing a red shirt, the total number is \(14\) because a green shirt was already selected and removed from the hamper.
Independent and Associated Events
An event is independent if the probability of it happening is not affected by another event. Two events are independent if the probability of each one occurring is not affected by the occurrence of the other.
When a die is thrown, that is an event. Its result is independent of other dice thrown before or after it. In a standard roulette wheel, the probability of a number appearing will always be \(1/38\), no matter the number of times the wheel is spun. Each spin is an independent event and not affected by the other spins.
Dependent events (also called associated events) refer to variables or events that have a relationship or connection. They may also be referred to as correlated variables. In the previous discussion of the shirts, picking the second shirt was an associated event because its probability was affected by the picking of the first shirt.
Relative Frequency
Like probability, relative frequency is a means of understanding how often a specific event happens in comparison to the total number of events. However, where probability is about the general possibility of an event happening, relative frequency measures actual occurrences. We use it when comparing groups or categories of data, such as survey results or experimental outcomes. In statistics questions on the SAT, relative frequency often appears in tables that show the relationship between two variables. For this exam, the key to interpreting data and recognizing patterns is understanding how to read the information as it is presented in tables.
Two-Way Table
Two-way tables are usually used to present survey results in tabular form. The columns (vertical) show the count or number for one category, while the rows (horizontal) indicate another category. The two-way table below shows the results of a survey conducted among \(125\) college students on the brand of beverage they prefer during lunch break.
| Brand A | Brand B | Brand C | |
|---|---|---|---|
| Male | \(42\) | \(5\) | \(18\) |
| Female | \(35\) | \(15\) | \(10\) |
| Total | \(77\) | \(20\) | \(28\) |
Many questions can be answered by finding the correct cell. For instance, if we want to know which brand is the least preferred by male students, we simply compare the three columns within the “Male” row. The table can also be used to compute answers that may not be readily provided.
Table 1: Frequency Table
| Brand A | Brand B | Brand C | Total | |
|---|---|---|---|---|
| Male | \(42\) | \(5\) | \(18\) | \(65\) |
| Female | \(35\) | \(15\) | \(10\) | \(60\) |
| Total | \(77\) | \(20\) | \(28\) | \(125\) |
There are different types of two-way tables. The table given above is known as a frequency table, so called because it shows the frequency or the count of an event occurring. In the context of the given example, the frequency is the number of times that a particular brand of beverage was chosen by the participating students. The numbers in the inner cells are called the frequencies or count.
The numbers in the Total column and Total row are called marginal frequencies, while those in the inner cells are called joint frequencies. Looking at the marginal frequencies alone, it’s clear Brand B is the least preferred beverage overall. Looking at the joint frequencies, however, we can see Brand C is the least favored beverage among female students.
Table 2: Relative Frequency Table
| Brand A | Brand B | Brand C | Total | |
|---|---|---|---|---|
| Male | \(42/125=0.34\) | \(5/125=0.04\) | \(18/125=0.14\) | \(0.52\) |
| Female | \(35/125=0.28\) | \(15/125=0.12\) | \(10/125=0.08\) | \(0.48\) |
| Total | \(77/125=0.62\) | \(20/125=0.16\) | \(28/125=0.22\) | \(1.00\) |
This data set can also be shown as a relative frequency table, such as the one above. It shows the frequency of an event occurring relative to the total number of events, hence, the term relative frequency. The relative frequencies or the decimal numbers in the inner cells are called conditional frequencies.
The conditional frequencies in Table 2 show the probability of a gender preferring a particular brand of beverage.
Note: We are showing the division of terms to illustrate the procedure, although relative frequency tables are normally shown with just the resulting decimal numbers.
Table 3: Relative Frequencies for Rows
| Brand A | Brand B | Brand C | Total | |
|---|---|---|---|---|
| Male | \(.34/.52=0.65\) | \(.04/.52=0.08\) | \(.14/.52=0.27\) | \(1.00\) |
| Female | \(.28/.48=0.58\) | \(.12/.48=0.25\) | \(.08/.48=0.17\) | \(1.00\) |
| Total | \(0.62\) | \(0.16\) | \(0.22\) | \(1.00\) |
Relative frequencies can be shown for the whole table, such as in Table 2. Relative frequencies may also be presented just for the rows and columns, as in Table 3 above. Or, they can be shown specifically in the columns, as with Table 4 below.
The conditional frequencies in Table 3 show the probability of each gender preferring a particular brand of beverage (e.g., the probability that male students will prefer Brand A is \(0.65\)).
Table 4: Relative Frequencies for Columns
| Brand A | Brand B | Brand C | Total | |
|---|---|---|---|---|
| Male | \(.34/.62=0.55\) | \(.04/.16=0.25\) | \(.14/.22=0.64\) | \(0.52\) |
| Female | \(.28/.62=0.45\) | \(.12/.16=0.50\) | \(.08/.22=0.36\) | \(0.48\) |
| Total | \(1.00\) | \(1.00\) | \(1.00\) | \(1.00\) |
The conditional frequencies in Table 4 show the probability of a brand being preferred by a particular gender (e.g., the probability that those who prefer Brand A will be male is \(0.55\)).
Probability Problems with Two-Way Tables
On the SAT, you should expect to see various questions that involve these types of two-way tables. The solutions are generally simple, but the questions need a little getting used to. Let’s practice a few examples.
Referring to Table 1 (above), what is the probability of randomly selecting a male participant who prefers Brand B?
Solution
This problem asks you to use a two-way table to calculate probability. While it might seem daunting to combine the two different topics, it’s actually quite straightforward if you remember the probability formula you learned earlier:
\[P(E) = \frac{n(E)}{n(\varepsilon)}\]where \(P(E)\) is the probability of a specific event happening, \({n(E)}\) is the desired outcomes, and \({n(\varepsilon)}\) is all possible outcomes.
Using the information from Table 1, we have two values: the number of male students who like Brand B and the total number of students of both genders. We can insert those values into our formula:
\[P(E) = 5 \div 125 = 0.04\]Therefore, the total probability of randomly selecting a male student who prefers Brand B is \(0.04\). Remember that probability is always a value between \(0\) and \(1\).
Note: \(0.04\) is a conditional frequency in the “Brand B” column of Table 2.
What is the probability of randomly selecting a participant who prefers Brand B, given that the participant is male?
At first glance, this may seem like the same question you just answered. However, the phrase “given that” in this type of question changes the meaning. It’s establishing a condition. In this case, the condition is that the student must be male. This means that the value for all “possible outcomes” is not \(125\), but rather \(65\) as that is the total number of male students. Therefore, our probability formula looks like this:
\[P(E) = 5 \div 65 \approx 0.08\]The \(\approx\) symbol means that we’ve rounded our answer. Notice, also, that this answer reflects what’s shown in the “Brand B” column in Table 3.
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