Calculating Probability
Probability is a numerical measure of how likely an event is to occur. It is usually denoted by P.
A simple event is often written as E (or Ei), and a compound event as A.
Two core properties of probability
1) Range property (0 to 1)
Whether the event is simple or compound, a probability can never be less than 0 and can never be greater than 1.
\[
\begin{aligned}
0 &\le P(E_i) \le 1 \\
0 &\le P(A) \le 1
\end{aligned}
\]
An impossible event has probability 0, while a sure (certain) event has probability 1.
\[
\begin{aligned}
P(\text{impossible}) &= 0 \\
P(\text{sure}) &= 1
\end{aligned}
\]
2) Sum-to-1 property for all simple outcomes
If an experiment has a complete list of mutually exclusive simple outcomes E1, E2, …,
then the total probability must be 1.
\[
\begin{aligned}
\sum_i P(E_i) &= P(E_1) + P(E_2) + P(E_3) + \cdots \\
&= 1
\end{aligned}
\]
Examples: one toss of a coin has P(H) + P(T) = 1; two tosses have
P(HH) + P(HT) + P(TH) + P(TT) = 1; a game can have
P(win) + P(loss) + P(tie) = 1.
Three conceptual approaches to probability
1) Classical probability (equally likely outcomes)
When all outcomes are equally likely, probability is computed by counting:
a simple event probability is 1 / N, and a compound event probability is
k / N, where N is the total number of outcomes and k is the number favorable to the event.
\[
\begin{aligned}
P(E_i) &= \frac{1}{N} \\
P(A) &= \frac{k}{N}
\end{aligned}
\]
Example idea: for a fair die, “even number” corresponds to outcomes \{2,4,6\}, so k = 3 and N = 6.
2) Relative frequency concept (sample data)
When outcomes are not equally likely (or when we rely on observed data), probability can be estimated by repeating an experiment
many times. If an event A is observed f times in n trials, then the relative frequency estimate is:
\[
\begin{aligned}
P(A) &\approx \frac{f}{n}
\end{aligned}
\]
The estimate tends to stabilize as n grows (an idea summarized by the Law of Large Numbers).
3) Subjective probability
In some situations there is no equally likely model and the experiment cannot be repeated in a controlled way.
In that case, a probability may be assigned based on judgment, experience, information, and belief.
Subjective probability is still written using the same notation P(A), but its value is based on informed assessment rather than counting or repeated trials.
How this calculator matches the lesson
- Classical mode: computes
P(A) = k / N for equally likely outcomes.
- Relative frequency mode: computes
P(A) \approx f / n using sample counts.
- Property checker: verifies the
[0, 1] range rule and whether a set of simple-event probabilities sums to 1.
- Visualization: displays
P on a 0-to-1 scale and animates the marker to the computed value.
