1. Probability and Statistics: Overview
====================================================

Industrial Engineering deals with systems that are not always easy to predict.
Measurements may contain errors, customer demand may change, and production
processes may not behave in exactly the same way every time.

Probability and statistics help us describe, study, and manage this uncertainty.

Modeling Uncertainty with Data
------------------------------

Uncertainty means that we do not know something exactly.

It may come from limited information, measurement errors, changing conditions,
complex systems, or unknown future events.

A system does not need to be completely random for us to be uncertain about it.
For example, a coin toss follows physical laws. Its result depends on force,
angle, spinning speed, gravity, air movement, and the landing surface. However,
we usually cannot measure and control all these factors exactly. Therefore, the
result appears random to us.

Data allow us to observe uncertain systems. Probability and statistics then help
us describe possible outcomes, identify patterns, and support decisions.

Measurement Uncertainty
~~~~~~~~~~~~~~~~~~~~~~~

A measurement is not always exact.

For example, repeated measurements of the same product may be slightly
different because of the instrument, operator, environment, or measurement
method.

.. math::

   \text{Measured value}
   =
   \text{true value}
   +
   \text{measurement error}

The true value may be fixed, but our knowledge of it is uncertain.

A key question is:

**Is the product really different, or is the difference caused by measurement error?**

Demand Uncertainty
~~~~~~~~~~~~~~~~~~

A company cannot know future demand exactly.

For example, a store may forecast that it will sell 100 units, but the actual
demand may be 80, 100, or 120 units.

.. math::

   \text{Actual demand}
   =
   \text{forecast}
   +
   \text{forecast error}

A key question is:

**How much inventory should the company prepare when future demand is uncertain?**

Process Uncertainty
~~~~~~~~~~~~~~~~~~~

Production and service systems are not always stable or predictable.

Customer arrivals, processing times, machine failures, transportation times,
and worker performance may change over time.

These outcomes usually have causes, but it may be difficult to observe and
model every cause.

A key question is:

**How much capacity is needed when operating conditions are uncertain?**

The Main Idea in Industrial Engineering
---------------------------------------

Industrial Engineering uses probability and statistics to:

* measure uncertainty,
* describe variation,
* learn from data,
* predict possible outcomes,
* evaluate risk, and
* make better decisions.

The goal is not always to remove uncertainty. In many real systems, this is not
possible.

The goal is to understand uncertainty and make good decisions even when the
information is incomplete.

.. math::

   \text{Measure uncertainty}
   \rightarrow
   \text{understand uncertainty}
   \rightarrow
   \text{make better decisions}

Probability
-----------

Probability starts with an assumed probability model or probability
distribution.

The model is known, but the future data or outcomes are not yet known.

.. math::

   \text{Known probability distribution}
   \rightarrow
   \text{unknown data or outcomes}

Probability asks:

**What outcomes may occur, and how likely are they?**

For example, if we assume that a coin is fair, then

.. math::

   P(\text{Heads}) = 0.5

Before the coin is flipped, the result is unknown. Probability describes the
chance of each possible result.

Statistics
----------

Statistics starts with observed data.

The data are known, but the underlying probability distribution, population,
or process is unknown.

.. math::

   \text{Known data}
   \rightarrow
   \text{unknown probability distribution or model}

Statistics asks:

**What can the observed data tell us about the population or process?**

For example, if a coin is flipped 100 times and produces 58 heads, statistics
can be used to estimate the probability of heads and evaluate whether the coin
may be fair.

The Main Difference
-------------------

Probability moves from a model to possible data.

.. math::

   \boxed{\text{Probability: model} \rightarrow \text{data}}

Statistics moves from observed data to an unknown model.

.. math::

   \boxed{\text{Statistics: data} \rightarrow \text{model}}

Probability describes uncertainty before an outcome is observed.

Statistics uses observed data to reduce uncertainty about an unknown system.