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Basics of Probability

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"CK-12 FlexBooks on Probability explains whether an event will or will not occur. Using the laws of probability, we can find the likelihood of two events occurring together, not occurring, or a large variety of other combinations."

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Study Guides

Big Picture

Probability is the study of whether an event will or will not occur. Using the laws of probability, we can find the likelihood

of two events occurring together, not occurring, or a large variety of other combinations.

Key Terms

Probability: A measure of how likely an event is.

Complementary Event: All the other events in a

Event: Something that occurs. Can have one or more

sample space.

possible outcomes.

Compound Event: An event made up of two or more

Simple Event: An event that has exactly one outcome.

simple events.

Sample Space: All the possible outcomes of a

Dependent Events: Events whose outcomes do affect


each other.

Outcome: The result of a single experiment.

Independent Events: Events whose outcomes don’t

Experiment: The process of taking a measurement or

affect each other.

making an observation.

Determining Probabilities

The probability of an event can be calculated by knowing the number of ways the event can occur and the size of

the sample space.

Probability of event A = P(A) =

Probability of an event is always between 0 and 1: 0 = impossible, 1 = always happens

The probabilities of all possible outcomes of an event must add up to 1. This means one of the outcomes must happen.

This method of determining probabilities assumes that all the possible outcomes are equally likely to happen.

Simple Events

Example: the probability that a die will land on 3

Experiment: rolling a single die

Event: the die lands on 3

This is also a simple event because the die can only land on one number (one possible outcome)

Sample space S lists the possible outcomes: S = {1, 2, 3, 4, 5, 6}

Size of sample space = six

P(die lands on 3) =

Non-Simple Events

Example: the probability that a die will land on either 2 or 3

Event: the die lands on 2 or 3

Can be broken down to two simple events: the die lands on 2, the die lands on 3

Sample space S = {1, 2, 3, 4, 5, 6}

Size of sample space = six

P(die lands on 2 or 3) =

There are two ways for the event to happen – the die can land on 2 or the die can land on 3



Complementary Events

Complement of an event A = A’ = all the events other than A in the sample space

Finding probabilities using complements:

P(A’) = probability that A doesn’t happen

Example: Throwing a die

Event A = observing an odd number, Event A’ = observing an even number

The Complement Rule:

P(A) + P(A’) = 1

can be rearranged: P(A’) = 1–P(A)

The Complement Rule is useful when P(A’) is easier to find than P(A)

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This guide was created by Lizhi Fan and Jin Yu. To learn more about the student

authors, visit





Compound Probabilities

The probability of compound events depend on the outcomes from two or more events.

Probability of both A and B happening: P(A and B) = (A B) = P(A) ∙ P(B, given A)

means the intersection of events

Probability of either A or B happening: P(A or B) = P(A

B) = P(A) + P(B)–P(A B)

means the union of events


Conditional Probabilities

Conditional probability: the probability of event A occurring, given event B has occurred

Notation: P(A|B), which means “the probability of A, given B”

P(A|B) =

Can rewrite as P(A B) = P(A|B) ∙ P(B)

This occurs when events are dependent, where the occurrence of the event will change the probability of the successive

events from occurring.

For example, taking a king randomly from a set of cards will decrease the probability of another king being taken


Independent Events

If an event is independent, its occurrence does not change the probability of successive trials from occurring.

For example, rolling a six on a die will not change the probability of rolling a six on the next die.

If A and B are independent events, then P(B|A) = P(B).

So P(A and B) = P(A) ∙ P(B, given A) = P(A) ∙ P(B) for independent events

Mutually Exclusive Events

If two events are mutually exclusive, meaning that they cannot occur together, then P(A B) = 0.

So P(A or B) = P(A

B) = P(A) + P(B) - P(A B) = P(A) + P(B)

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Tree Diagrams & Counting Techniques

Tree diagrams are useful for showing all the possible

outcomes when there is a series of events.

Example: A box contains three balls: a red (R) ball, a blue

(B) ball, and a white (W) ball. One ball is selected and

then returned back into the box. The balls are scrambled

and another ball is selected. The tree diagram shows the

sample space:

There are 9 possible outcomes.

You can also calculate the number of possible outcomes

by using the multiplication rule of counting:

If there are n possible outcomes for event A and m

for event B, there is n • m possible outcomes for

event A followed by event B.

Combination and Permutation

Counting Rule for Combinations

The number of combinations of n objects taken r at a time is:

For example,

C6 is the number of combinations of six objects that can be chosen from 13 objects.


Combination = order does not matter

To remember that order does not matter for combinations, imagine a combo pizza where you choose whatever

toppings you want and just throw it on!

Counting Rule for Permutations

The number of ways to arrange n objects in order within r positions is:

For example,

P6 is the number of ordered ways to arrange six objects chosen from 13 objects.


Permutation = order does matter

For both rules, n and r must be whole numbers, and n ≥ r.


Factorial notation is a shorthand way to write out a common multiplication pattern in probability and statistics:

n! = n(n-1)(n-2)(n-3)...(3)(2)(1)

The ! is the symbol for factorial. n! is the n factorial.

Law of Large Numbers

The law of large numbers simply says that the more an experiment is repeated, the closer the results will get to the

theoretical value.

For example, say you flip a coin 3 times and get two heads and one tails. The proportion of getting a head seems to


while getting a tail seems like .

However, we know that the true value of getting heads or tails should be . If you flip the coin more, the probabilities

will even out as the number of trials increase. The probabilities will get closer and closer to chance of getting heads

and chance of getting tails.

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