Probability

Probability measures how likely it is that an event will occur. Probability values range from zero, meaning the event will definitely not happen, to one, indicating that the event is guaranteed to happen.

Rules

When computing probabilities involving 2 events, A and B, there are two possibilities:

• The events are independent. In that case the probability of both events occurring, \(P(A \text{ and } B)\) equals the product of the individual probabilities, \(P(A) \cdot P(B)\).

• The events are dependent. In this case \(P(A \text{ and } B)\) is \(P(B) \cdot P(A\vert B)\), where \(P(A \vert B)\) is the probability of event A given that event B has already occurred. This is conditional probability.

Conditional Probability and Independence

If you already know the probabilities for two events A and B occurring both separately and together, you can analyze these probabilities to determine if the events are independent. If you see that \(P(A \text{ and } B)\) equals \(P(A) \cdot P(B)\), then the events are independent. Similarly, if you see that \(P(A)\) and \(P(A \vert B)\) are the same, this means that \(P(A)\) is the same whether or not B occurs, again showing that the events are independent.

Subsets

When considering an event, for example rolling a die twice, the set of all possible outcomes is the sample space (in this case, \(6\cdot 6 = 36\) possible outcomes). Each specific outcome is a subset of the sample space. Rolling a 2 and a 5 is the intersection of the two subsets, \(P(A \text{ and } B) = P(A) \cdot P(B)\). Rolling a 2 or a 5 is the union of the two subsets, \(P(A \text{ or } B) = P(A) + P(B)\). Rolling a 2 and not rolling a 2 are complementary events, \(P(A) + P(B) = 1\).

Two-Way Frequency Tables

Two-way frequency tables can be used to estimate conditional probabilities and assess whether events are independent. For example, in a school survey reporting students’ grade level and favorite subject, the probability that the favorite subject is science given that they are sophomores is (# of sophomores liking science)/(# of sophomores). The probability that any student likes science is (# of students liking science)/(total # of students). If these probabilities are close to equal then liking science and being a sophomore are independent.

Explaining Probability

In this test you will need to be able to recognize independent and conditional probabilities in everyday situations and to explain these concepts in everyday language. For example, the chance of having lung cancer if you are a smoker may be different than the chance of being a smoker if you have lung cancer. In the first case, you are comparing against all smokers and divide by the number of smokers. In the second case you are comparing against all people with lung cancer and divide by the number of people with lung cancer.

Other Math Abilities Tested

There are additional skills and concepts tested on the SBAC Mathematics test. Be sure to go over them here at the end of the last page of our SBAC High School Mathematics: Numbers and Operations Study Guide

Tackling Differently Formatted Test Items

There is important information about differently formatted test items on the SBAC exam. Go to our home page for the SBAC to read it as you prepare. Scroll down to “Tips and Tricks.”