We live
our lives by the calculus of chance, often without even realizing it. We
casually state, “It probably won't rain,” or “I’ll likely change jobs soon.”
These everyday phrases reveal an intuitive grasp of probability. But beyond
this subconscious use, we also engage in conscious, deliberate calculation.
Ask
someone about the probability of a coin landing on “heads,” and the answer
comes quickly: 1/2 or 50%. Why? Because there are two possible outcomes—heads
or tails—each equally likely. Therefore, the probability of heads is 1/2.
🎲 Events
and Sample Space
At the
heart of any probabilistic phenomenon lies an event—a single
outcome. The set of all possible events is called the sample space.
Example:
When you roll a fair die, the sample space consists of the six possible
results. Each face represents a distinct event: 1, 2, 3, 4, 5, and 6.
📘 The
Classical Definition
The
probability of an event A occurring is defined as the ratio of the number of
ways A can happen to the total number of all possible outcomes, all under
identical conditions.
- The
probability of event A is denoted as P(A).
- The
sum of the probabilities of all events within a single sample space is
always equal to 1.
- By
definition, the probability of any event is a number between 0 and 1.
This
classical, or frequentist, approach is the most intuitive. It
applies perfectly to repeatable phenomena where we can observe many occurrences
under the same conditions.
Example:
A doctor finds that out of 2,964 live births, 73 infants had a serious birth
defect or condition. The estimated probability of a newborn presenting with one
of these conditions is:
P(A) = 73 / 2964 ≈
0.0246
⚖️ Probability and Risk
In
healthcare, the probabilities of adverse events are often termed risks.
Example:
A study analyzing 30,195 hospital records identified 1,133 cases of serious
injury caused by medical error. The estimated risk of serious injury in that
hospital was:
P(A) = 1133 / 30195
≈ 0.0375
🧩 The
Limits of the Classical View
The
frequentist definition works well when the number of observations can grow
indefinitely. However, it falls short in situations where this isn't feasible.
Example:
Stating that “The probability of Brazil winning the next World Cup is 0.95”
does not fit the frequentist mold. For such one-off or uncertain future events,
we turn to the subjective definition of probability.
💭
Subjective Probability
Subjective
probability is a value between 0 and 1 that expresses a personal degree
of belief in the occurrence of an event. It is not based on formal
calculation but on knowledge, experience, and rational judgment.
- It
is invaluable when information is scarce, yet a decision must be made.
- This
approach is common in clinical, financial, and managerial decisions,
where informed intuition plays a crucial role.
- Its
main limitation is its personal nature—two individuals may assign
different probabilities to the same event, and only repeated observation
(if possible) can reveal whose belief was better calibrated to reality.
🔢
Decimals vs. Percentages
Statisticians
prefer to express probabilities as numbers between 0 and 1, as this notation is
essential for more complex calculations. However, for the general public,
expressing them as percentages is often more intuitive,
achieved simply by multiplying the decimal value by 100.
Example:
If a hospital has 120 beds and 87 are occupied, the occupancy rate is:
P(A) = 87 / 120 =
0.725
Therefore, the occupancy percentage is 72.5%.
Conclusion
From the
most trivial decision to the most complex scientific prediction, probability is
the tool that allows us to navigate an uncertain reality. Understanding its
definitions—whether the classic one, which measures frequencies, or the
subjective one, which quantifies our beliefs—is not just an academic exercise.
It is a way for us to make more informed decisions and view the world with a
more critical and enlightened eye.
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