Probability

Learning Objectives

• Calculate the probability of a simple event as (favorable outcomes) / (total outcomes) and express the result as a fraction, decimal, or percent
• Apply the complement rule: P(event does not occur) = 1 - P(event occurs)
• Calculate probability for independent events by multiplying: P(A and B) = P(A) x P(B) when A and B are independent
• Identify dependent events and adjust the probability of the second event based on the outcome of the first
• Apply the addition rule for mutually exclusive events: P(A or B) = P(A) + P(B)
• Apply the addition rule for overlapping events: P(A or B) = P(A) + P(B) - P(A and B)
• Calculate conditional probability from a two-way table by using the specified subgroup as the denominator
• Apply the counting principle: if event 1 can occur in m ways and event 2 can occur in n ways, then both can occur in m x n ways

High-Yield Concepts

Study Strategy

Start with the basic probability formula: P(event) = favorable outcomes divided by total outcomes. Practice identifying what counts as a favorable outcome and what constitutes the sample space (total outcomes). The complement rule (1 minus the probability of the event) is often faster than computing the event directly when the event is complex.

Then study independent versus dependent events. Independent events do not affect each other -- flipping a coin twice produces independent outcomes. Multiply probabilities for independent "and" problems. Dependent events do interact -- drawing two cards without replacement changes the composition of the deck. For dependent events, the probability of the second event must account for what happened first.

Study mutually exclusive versus overlapping events. Two events are mutually exclusive if they cannot both occur at the same time (like rolling a 3 and rolling an even number on a single die roll). For mutually exclusive events, P(A or B) = P(A) + P(B). For overlapping events (which can both occur simultaneously), you must subtract P(A and B) to avoid double-counting.

Conditional probability is the most conceptually tested topic in this unit. P(A given B) restricts the sample space to the cases where B occurred. From a two-way table, this means using the row or column total for the condition as the denominator. Practice identifying which total is the "given" condition.

For counting problems, apply the fundamental counting principle: multiply the number of options at each step. Combinations (order does not matter) and permutations (order matters) are less frequently tested but appear for simple cases.

Common Mistakes

Adding probabilities for independent "and" problems. P(rolling a 3 on the first die AND rolling a 3 on the second die) = (1/6) x (1/6) = 1/36, not 1/6 + 1/6. For independent events that must both occur, multiply the probabilities.

Forgetting to subtract the overlap in "or" problems. If a class has 15 students who play soccer and 12 who play basketball, and 5 play both, then the number who play soccer OR basketball = 15 + 12 - 5 = 22. Failing to subtract the overlap double-counts the 5 students who play both sports.

Not adjusting for dependent events. When drawing without replacement, the second draw depends on the first. If 3 red and 7 blue balls are in a bag, and one red is drawn first, the probability the second draw is also red is 2/9 (2 red remain out of 9 total), not 3/10.

Using the wrong denominator for conditional probability from a two-way table. "What is the probability that a randomly selected student who passed is female?" uses the total number of students who passed as the denominator, not the total number of female students or the grand total.

Confusing "at least one" with simple probability. P(at least one) problems are best solved using the complement: P(at least one) = 1 - P(none). Computing P(at least one) directly requires adding many cases; using the complement is almost always faster.

Exam Tips

Geometric probability problems present a region within a larger region and ask for the probability that a randomly placed point falls in the inner region. The probability equals the area of the target region divided by the area of the total region. These problems combine probability with geometry (area formulas for rectangles, circles, triangles). Always compute both areas before dividing.