Probability of a Single Event
In basic probability, we begin with a sample space \(S\), which is the set of all possible outcomes of an experiment.
An event \(A\) is any subset of \(S\). For example, when rolling a fair die once, the sample space is
\(S=\{1,2,3,4,5,6\}\). The event “roll an even number” is \(A=\{2,4,6\}\).
If all outcomes are equally likely (the uniform case), the probability of event \(A\) is computed by counting:
\[
P(A)=\frac{|A|}{|S|},
\]
where \(|A|\) is the number of favorable outcomes and \(|S|\) is the total number of outcomes. This “counting formula” is why many
introductory problems ask for “favorable over total.” For a standard deck of 52 cards, the event “draw an ace” has \(|A|=4\) favorable
outcomes and \(|S|=52\), so \(P(A)=4/52=1/13\).
Probability is often reported in multiple formats. A fraction is exact, while a decimal and percent are convenient for interpretation:
\(0.0769\) means about \(7.69\%\). Converting is straightforward:
\[
\text{percent} = 100 \times P(A).
\]
When possible, simplifying the fraction (dividing numerator and denominator by their greatest common divisor) makes the result clearer.
Not every situation is uniform. In a non-uniform (weighted) model, outcomes can have different probabilities.
A common way to represent this is with weights \(w_i\ge 0\) assigned to each outcome \(i\in S\). The probability of an event
becomes:
\[
P(A)=\frac{\sum_{i\in A} w_i}{\sum_{i\in S} w_i}.
\]
If the weights already sum to \(1\), they are literal probabilities. Otherwise, the formula normalizes them automatically.
This approach models loaded dice, biased coins, or any discrete experiment where outcomes are not equally likely.
Two useful related ideas are the complement and the impossibility/certainty bounds. The complement of \(A\),
written \(A^c\), contains all outcomes not in \(A\). Since an outcome must be either in \(A\) or not in \(A\),
\[
P(A^c)=1-P(A).
\]
Also, every probability satisfies \(0\le P(A)\le 1\): an impossible event has probability \(0\), and a certain event has probability \(1\).
This calculator focuses on single-event probabilities in finite sample spaces. Once you’re comfortable with these foundations,
you can extend to combined events (\(A\cup B\), \(A\cap B\)), conditional probability, and independence.
