Probability

Probability theory is a branch of mathematics concerned with the science of chance. Probability describes the real world. The data we collect and models we build involve statistics and statistical learning.

Basic terminology

A random experiment is one where there’s a varied outcome each time. We define the sample space $S$ as the set of all possible outcomes. We define the likelihood of each outcome in $n$ trials as $N_k(n)$, and the relative frequency of an event $k$ in $n$ trials as $f_k(n)=\frac{N_k(n)}{n}$, such that the probability of an event $k$ is:

$$P(k)=P_k=\underset{n\to \infty }{\lim }{f}_k(n)$$

In practice, an infinite number of trials is impossible, so we instead define several axioms to model probability:

• The probability of an event $A$, $P(A)\in [0,1]$.
• Total probability, i.e., of the sample space, is $P(S)=1$.
• If events $A,B$ are mutually exclusive, then $P(A\cup B)=P(A)+P(B)$.

We have some handy formulas, but it’s nice to first define some terms.

• An event is a condition on an outcome. For example, an event $A$ such that a roulette result is even and is non-zero.
  • If an event produces one outcome (instead of a subset of $S$), then we call this an elementary event.
  • Complicated events are a set union of elementary events.
• The complement of an event $A$ is the event that $A$ does not occur. Complementary events are mutually exclusive and exhaustive events; i.e., their probabilities will together sum to 1.
  • We additionally define the complement of $A$ as $A^{\prime }$ or $\bar{A}$ or $A^C$. Then, $\bar{A}=S\backslash A$.
• Mutual exclusivity of two events suggest that they cannot occur at the same time.
• Events are exhaustive if at least one event must occur, i.e., when rolling a six-sided die.
• Events are independent if one event occurring does not affect the probability of another event occurring.

And some useful axioms:

• For complementary events, $P(A)+P(A^{\prime })=1=S$.
• For combined events, $P(A\cup B)=P(A)+P(B)-P(A\cap B)$.
• For mutually exclusive events, $P(A\cup B)=P(A)+P(B)$. In fact, if $S$ is the union of $k$ distinct mutually disjoint subsets, then $P(S)$ is the sum of the probability of each subset.
• For conditional events, $P(A|B)=\frac{A\cap B}{P(B)}$.
  • Conditional: probability of event $A$ occurring given that event $B$ has occurred.
• For independent events, $P(A\cap B)=P(A)P(B)$.

A random variable assigns a numerical value to each outcome in a sample space. A random variable is discrete if its possible values form a discrete set. Continuous random variables contains an interval where all points exist, i.e., its probabilities are given by areas under a curve.

Extensions

It also helps us to define a few formulas given certain situations. If every outcome is equally likely, then the probability of an event $E$ is:

$$P(E)=\frac{\text{number of outcomes in }E}{\text{total outcomes in}S}$$

Constructing statements for the probability of certain events often takes combinatorial approaches, including permutations and combinations.

For situations that involves us successively making decisions (like sampling from a set of items more than once). In this case, we can refer to the multiplication rule.¹ This holds regardless of how the preceding steps were formed:

If an operation consists of $k$ steps, and the first step can be performed in $n_1$ ways, second in $n_2$ ways, $k$th step in $n_k$ ways, then the entire operation is performed in: $n_1{n}_2\cdots n_k$.

From here, we can obviously tell that if $n_1=n_2=n_k$, then the operation has $n^k$ possibilities. If we don’t replace an element in the sample space (i.e., we can’t repeat an element), then we have $n_1=n_2+1$, and the operation has $n!$ possibilities.

Sub-pages

• Bayes’ theorem
• Probability distribution

Footnotes

1. This name was pulled from Discrete Mathematics with Applications, by S.S. Epp. ↩