Probability—describes the likelihood of a certain event; all probability values fall between 0% and 100%, or between 0 and 1, inclusive.

Probability of a single event: P(A) =

Mutually exclusive events—events that can never occur together (i.e., the occurrence of one event completely eliminates the probability of occurrence of the other).

Examples:

Getting an A, a B, and a C as the final grade on one class

Answering 17 questions right and answering 21 questions right on a particular exam

Important Properties of Mutually Exclusive Events

1. Mutually exclusive events never occur together: P(A and B) = 0.

2. If A and B are mutually exclusive, then P(A or B) = P(A) + P(B).

Two events are complementary if one and only one of them must occur. All complementary events are mutually exclusive (because only one of them can occur), but not all mutually exclusive events are complementary.

Examples:

Failing or passing a certain course (contrast with getting a particular grade)

Getting at least one head on three flips of a coin or getting no heads

Important Properties of Complementary Events

1. Complementary events never occur together: P(A and B) = 0.

2. Because one of the complementary events must occur, their probabilities sum up to 1: P(A or B) = 1; P(A) = 1 – P(B); P(B) = 1 – P(A).

Events A and B are dependent if the occurrence of one event affects the probability of another.

Examples:

Drawing balls without replacement

Allocating a limited number of prizes among the audience (Each subsequent allocation of a prize reduces the probability of the remaining participants winning.)

Important Properties of Dependent Events

1. P(A and B) = P(A) • PA(B)

Where P(A and B) is the probability that both events will occur; P(A) is the probability of event A; PA(B) is the probability that event B will occur assuming that A has already occurred; PA(B) is called conditional probability.

Events A and B are independent if the occurrence of one event does not affect the probability of the occurrence of another. In case of sequential events, the initial situation is restored before each subsequent experiment.

Examples:

Tossing a coin

Selecting balls from a jar with replacement

Important Properties of Independent Events

• P(A and B) = P(A) • P(B)

OR formulas (probability of A or B):

• General case: P(A or B) = P(A) + P(B) – P(A and B)

• Mutually exclusive events: P(A or B) = P(A) + P(B)

• Complementary events: P(A or B) = 1

AND formulas (probability of A and B):

• Mutually exclusive and complementary events: P(A and B) = 0

• Dependent events: P(A and B) = P(A) • PA(B)

• Independent events: P(A and B) = P(A) • P(B)

Other Important Strategies:

• Always consider if you are dealing with dependent probability. It is easy to miss dependent probability on questions.

• Often it is easier to find the complementary probability, particularly when you see the words “at least one.” If a question seems too complicated, it may be because you can easily solve for the complementary probability.

• Read carefully, particularly in binomial probability questions. There is a big difference, for instance, between asking the probability of getting “two heads in a row and then a tail” and “two heads and a tail in any order.”

• Many probability questions are best solved using combinatorics or writing out the possibilities. Probability is not always multiplying or adding probabilities; sometimes it is just figuring out the number of favorable outcomes and the number of total outcomes.

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