Probability: Cheat Sheet

Download a pdf of this page here.

Essential Concepts

  • In experiments, the result is called an outcome, and events can be specific results or groups of results. Simple events are indivisible outcomes, while compound events are combinations of two or more simple events. The sample space refers to the set of all possible simple events that can occur.
  • In basic probability, if all outcomes have an equal chance of happening, we can find the probability of an event by dividing the number of outcomes for that event by the total number of equally likely outcomes. Probabilities can be written as decimals, fractions, or percentages, and we use the notation “[latex]P(event)[/latex]” to represent the probability of an event.
    • An impossible event has a probability of [latex]0[/latex].
    • A certain event has a probability of [latex]1[/latex].
    • The probability of any event must be [latex]0 ≤ P(E) ≤ 1[/latex].
  • Complementary events are the opposite of each other. The complement of an event “[latex]E[/latex]” is when “[latex]E[/latex]” doesn’t happen.
    • [latex]P\left({\bar{E}}\right)=1-P(E)[/latex]
  • If events A and B are independent, it means that whether event A happens or not doesn’t affect the probability of event B happening. In other words, the chances of event B occurring are the same regardless of whether event A occurs or not.
  • The probability of independent events
    • [latex]P\left(A\text{ and }B\right)=P\left(A\right)\cdot{P}\left(B\right)[/latex]
    • [latex]P(A\text{ or }B)=P(A)+P(B)–P(A\text{ and }B)[/latex]
  • Conditional probability deals with events that are dependent on each other, unlike the independent events we’ve discussed before. It means that the probability of event B happening depends on whether event A has already happened.
    • We represent the probability of B given A as [latex]P(B|A)[/latex]
    • If the events A and B are not independent, then [latex]P(A \text{ and } B) = P(A) · P(B | A)[/latex]
  • Bayes’ Theorem is a mathematical formula used to update probabilities based on new information. It helps us calculate the probability of an event happening, given that we have some prior knowledge or evidence.
    • [latex]P(A|B)=\frac{P(A)P(B|A)}{P(A)P(B|A)+P(\bar{A})P(B|\bar{A})}[/latex]
  • The basic counting rule tells us how many choices we have when picking one item from each of two different categories. If there are “m” choices in the first category and “n” choices in the second category, then the total number of possible choices is the product of m and n, which is m multiplied by n. This rule is also known as the multiplication rule for probabilities.
  • Factorials are a way of multiplying a number by all the positive whole numbers that are smaller than it. Factorials are represented by an exclamation mark (!) after a number, and to find the factorial of a number, we multiply that number by all the positive whole numbers smaller than it, going down to [latex]1[/latex].
    • [latex]n! = n \cdot (n-1)\cdot (n-2)...1[/latex]
  • Permutation is a way of arranging objects in a specific order. In permutations, the order of the objects matters. We use the notation “P(n, r)” to represent the number of permutations of n objects taken r at a time.
    • [latex]P(n, r) = \frac{n!}{(n-r)!}[/latex]
  • Combinations are similar to permutations, but in combinations, the order of objects doesn’t matter. A combination is a selection of objects from a set without considering the order. The number of combinations of n objects taken r at a time is denoted by C(n, r).
    • [latex]C(n, r) = \frac{n!}{r!(n-r)!}[/latex]
  • Expected value is a really useful concept in probability. It helps us figure out the average gain or loss we can expect if we repeat an event many times. To find the expected value, we multiply each outcome by its probability, and then add up all those products.
    • [latex]E(x)=\sum_{i}^{n}P(x_i)x_i[/latex]

Glossary

basic counting rule

the total number of available choices is [latex]m \cdot n[/latex] where there are [latex]m[/latex] items in the first category and [latex]n[/latex] items in the second category

certain event

probability of [latex]1[/latex]

combination

a selection of objects from a set without regard to the order in which they are selected

complement of an event

the event “[latex]E[/latex] doesn’t happen”

compound event

a combination of two or more simple events

conditional probability

events that are dependent on each other

event

any particular outcome or group of outcomes

expected value

the average gain or loss of an event if the procedure is repeated many times

impossible event

a probability of [latex]0[/latex]

outcome

result of an experiment

permutation

an arrangement of a set of objects in a particular order. In permutations, the order in which the objects are arranged is important

sample space

the set of all possible simple events

simple event

an event that cannot be broken down further

Key Equations

Basic Probability

[latex]P(E)=\frac{\text{Number of outcomes corresponding to the event E}}{\text{Total number of equally-likely outcomes}}[/latex]

Bayes’ Theorem

[latex]P(A|B)=\frac{P(A)P(B|A)}{P(A)P(B|A)+P(\bar{A})P(B|\bar{A})}[/latex]

combination

[latex]C(n, r) = \frac{n!}{r!(n-r)!}[/latex]

complement

[latex]P\left({\bar{E}}\right)=1-P(E)[/latex]

expected value

[latex]E(x)=\sum_{i}^{n}P(x_i)x_i[/latex]

factorial

[latex]n! = n \cdot (n-1)\cdot (n-2)...1[/latex]

[latex]P(A \text{ and } B)[/latex] for Independent Events

[latex]P\left(A\text{ and }B\right)=P\left(A\right)\cdot{P}\left(B\right)[/latex]

[latex]P(A \text{ or } B)[/latex] for Independent Events

[latex]P(A\text{ or }B)=P(A)+P(B)–P(A\text{ and }B)[/latex]

permutation

[latex]P(n, r) = \frac{n!}{(n-r)!}[/latex]