Probability Theory Axioms and Distributions Cheat Sheet

Probability theory gives the rules for measuring uncertainty in statistics, data science, and scientific modeling. This cheat sheet summarizes the axioms, set operations, conditional probability, independence, expected value, and variance. College students need these ideas to derive results, choose distributions, and interpret probabilistic models correctly.

It is designed as a formula-forward reference for homework, exams, and applied analysis.

The foundation is that probabilities are nonnegative, total probability equals $1$ , and disjoint events add. Conditional probability uses $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$ to update probabilities when information is known. Moments such as $E[X]$ and $\operatorname{Var}(X)$ describe the center and spread of random variables.

Common distributions such as Bernoulli, Binomial, Poisson, Uniform, Exponential, and Normal connect these rules to real data models.

Key Facts

The probability axioms are $P(A) \ge 0$ , $P(S) = 1$ , and if $A_i$ are disjoint, then $P\left(\bigcup_i A_i\right) = \sum_i P(A_i)$ .
The complement rule is $P(A^c) = 1 - P(A)$ .
The addition rule for two events is $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ .
Conditional probability is defined by $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$ when $P(B) > 0$ .
Events $A$ and $B$ are independent if $P(A \cap B) = P(A)P(B)$ , equivalently $P(A \mid B) = P(A)$ when $P(B) > 0$ .
Expected value and variance are $E[X] = \sum_x x p(x)$ for a discrete random variable and $\operatorname{Var}(X) = E[X^2] - (E[X])^2$ .
For a Binomial random variable $X \sim \operatorname{Bin}(n,p)$ , $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$ , $E[X] = np$ , and $\operatorname{Var}(X) = np(1-p)$ .
For a Normal random variable $X \sim N(\mu,\sigma^2)$ , the standardized variable is $Z = \frac{X - \mu}{\sigma}$ .

Vocabulary

Sample space: The sample space $S$ is the set of all possible outcomes of a random experiment.
Event: An event $A$ is a subset of the sample space whose probability $P(A)$ can be measured.
Conditional probability: Conditional probability $P(A \mid B)$ is the probability that event $A$ occurs given that event $B$ has occurred.
Independence: Two events are independent when knowing one occurred does not change the probability of the other.
Expected value: The expected value $E[X]$ is the long-run average value of a random variable.
Probability distribution: A probability distribution assigns probabilities to possible values of a random variable while totaling $1$ .

Common Mistakes to Avoid

Adding overlapping events without subtracting the intersection is wrong because $P(A) + P(B)$ counts outcomes in $A \cap B$ twice.
Treating independent and disjoint events as the same is wrong because disjoint events with positive probabilities cannot be independent.
Using $P(A \mid B) = \frac{P(A)}{P(B)}$ is wrong because conditional probability must use the joint probability $P(A \cap B)$ in the numerator.
Forgetting distribution conditions is wrong because formulas such as $X \sim \operatorname{Bin}(n,p)$ require a fixed number of independent trials with the same success probability.
Confusing variance and standard deviation is wrong because variance is measured in squared units while standard deviation is $\sigma = \sqrt{\operatorname{Var}(X)}$ .

Practice Questions

1 If $P(A)=0.45$ , $P(B)=0.30$ , and $P(A \cap B)=0.12$ , find $P(A \cup B)$ .
2 If $P(A \cap B)=0.18$ and $P(B)=0.60$ , find $P(A \mid B)$ .
3 Let $X \sim \operatorname{Bin}(10,0.4)$ . Find $P(X=3)$ , $E[X]$ , and $\operatorname{Var}(X)$ .
4 Explain why two events with $P(A)>0$ , $P(B)>0$ , and $A \cap B = \varnothing$ cannot be independent.

Understanding Probability Theory Axioms and Distributions

A probability model starts by defining what outcomes are possible before any calculation begins. This full collection is the sample space. Events are groups of outcomes inside it.

Clear event definitions prevent many mistakes. For a card draw, getting a heart and getting a face card overlap because some cards meet both descriptions. Adding their probabilities without accounting for the overlap counts those cards twice.

By contrast, mutually exclusive events cannot occur on the same trial. Independence means something different.

Two independent events can occur together, but learning that one happened does not change the chance of the other. Students often confuse these ideas because both involve pairs of events.

Conditional probability is the language of evidence. Once a condition is known, the relevant sample space becomes smaller. Medical screening gives a useful example.

A test may be accurate for people who have a disease, yet a positive result may still not mean disease is likely when the disease is rare. The starting frequency in the population matters. This is called the base rate.

Bayes' rule organizes this reversal from the chance of a positive test given disease to the chance of disease given a positive test. Tree diagrams and two way tables are often safer than memorizing a rule because they show which group each probability belongs to.

A random variable converts outcomes into numbers that can be analyzed. Discrete variables count values, such as the number of defective items in a box. Continuous variables measure values, such as travel time or height.

For a continuous distribution, the probability at one exact value is zero. Probability comes from an interval, represented by area under a density curve. This explains why a density can sometimes be greater than one without breaking any rule.

Expected value is a long run average across many repetitions, not a promised result from one trial. Variance measures how far values tend to spread around that average. Large variance means outcomes are less predictable, even when two variables have the same mean.

Choosing a distribution depends on the process that generated the data. A binomial model fits a fixed number of trials when each trial has two outcomes, the success chance stays constant, and trials are independent. A Poisson model often fits counts of events in a fixed time or space interval, such as calls arriving each minute, when events occur independently at a roughly steady rate.

An exponential model describes waiting time between such events. The normal distribution is useful for measurements shaped by many small influences and for approximating some large counts. Before applying any model, check its assumptions.

Real observations can be dependent, skewed, limited by a maximum, or affected by changing conditions. A correct formula applied to an unsuitable model still gives a misleading answer.

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Understanding Probability Theory Axioms and Distributions