The concept of probability

This topic explores probability theory: its history, objectives, and key concepts crucial for our future studies.

From Chaos to Calculation: Exploring Probability

Mathematics plays a central role in natural sciences and continually evolves with our understanding of the world. Yet, the world is inherently unpredictable. Perfect predictions are often challenging due to multiple potential outcomes. For instance, despite technological advancement, weather forecasts have an element of uncertainty resulting in probabilities. It's the same with predicting a baby's gender. While some cultures have beliefs to influence it, the outcome is primarily due to chance. Humans naturally desire certainty, leading them to seek ways to manage uncertainty.

The $17$th-century birth of probability theory stemmed from mathematicians' interest in gambling, where they discovered probabilistic patterns in games like dice. These games proved ideal for studying simple random events.

To understand probabilities numerically, you need to grasp the terminology. How do we define and measure probability? How does it work in weather forecasts, or when selecting colors in roulette? Whether in marketing research or elsewhere, everything starts with the basics, and these basics apply to most random events.

Calculating probability

Early probability theorists agreed on calculating probabilities as the number of successful outcomes divided by the number of trials. For a coin toss, if heads occurred '$m$' times out of '$n$' tosses, the probability of heads is '$m/n$'. The more trials, the closer 'm' gets to $1/2$.

Statistics play a significant role in probability theory. Precise probabilities require many trials. For familiar objects like coins and dice, $17$th-century mathematicians already laid the groundwork. However, specialized events may need particular statistics.

For example, you might encounter 'manufacturing defect percentages.' This is also a probability: the count of defective items divided by the total produced. If out of $1000$ items, $7$ are defective and $993$ are not, the likelihood of receiving a defective item is $7/1000$ or $0.7\%$. Is this common? For technically complex items, a $2\%$ defect rate is considered standard, equivalent to $2$ faulty items out of $100$ or $2/100$. Extrapolated to $1000$, it becomes $20/1000$. So, $7/1000 < 20/1000$. In high-tech production, a $7/1000$ defect rate is commendable!

In some cases, calculating probability isn't straightforward and depends on indirect factors. Probability isn't always a simple $m/n$ ratio. For instance, what is the probability of encountering a dinosaur on the street? You could whimsically say $1/2$ as you either encounter one or you don't. However, given the numerous factors restricting a dinosaur's appearance on the street, including extinction, it's safe to assume this probability would be much lower. Keep in mind that numerous factors can impact an event's probability, and not all outcomes are equally probable or easily measured.

Some situations aren't repetitive, necessitating the consideration of other factors. We'll learn how to handle such cases in later topics. For now, let's familiarize ourselves with the basics.

Random, certain, and impossible events

Imagine a group of students contemplating their plans before an exam. One suggests, 'Let's flip a coin. Heads, we go to a bar; tails, we visit the girls; it lands on its edge, we go to sleep; it hangs in the air, we study for the exam.' This humorous example illustrates how random events can guide decision making. The outcome of a random event stays uncertain until unfolded. In the coin toss, we can predict potential outcomes.

An event reflects an experiment's outcome. In our example, there are four potential outcomes:

1. Heads

2. Tails

3. The coin lands on its edge.

4. The coin hangs in the air.

These outcomes don't share equal probabilities. For instance, how often have you seen a coin landing on its edge? It's even less likely to hang mid-air. Hence, option $4$ is an impossible event. Discussing a mathematical coin toss model, event $3$ is also impossible. It seems the students don't really want to study! On the other hand, certain events are bound to happen, such as the coin landing due to gravity. Whether it lands heads or tails is a random event with two potential outcomes. Events $1$ and $2$ are mutually exclusive; they can't occur simultaneously.

Let's introduce some math. Probability is a numeric expression, but what number? There's a fundamental rule:

• The probability of a certain event is $1$.

• The probability of an impossible event is $0$.

• The sum of all mutually exclusive outcomes of a certain event equals $1$.

For a coin toss: the coin will certainly land, hence its landing probability is $1$. But how will it land? Heads or tails are two random outcomes, each as likely as the other. Their combined probability equals $1$, making each outcome's probability $1/2$.

This same principle applies to other random events. For instance, when rolling a six-sided die, the probability of each face coming up is $1/6$, and for a $20$-sided die, it's $1/20$. The act of moving the die is a certain event, and the probability of each outcome depends on the total number of possible outcomes. Mathematicians from the $17$th century gathered data and showed that all these outcomes are equally probable. We will learn how to work with more complicated cases in further topics.

Conclusion

What should you remember from this topic?

• There are three types of events: random, certain, and impossible.

• Random events are often subsets of certain events.

• The sum of probabilities of all random outcomes in a certain event equals $1$.

• The probability for a random event falls between $0$ and $1$, with higher values indicating greater frequency (determined through statistics).

• Some events have a set of mutually exclusive and equally probable outcomes in which probabilities can be calculated easily; others request more complicated calculations and suggestions.

• The probability of an impossible event is $0$.