Probability is a way to measure how likely something is to happen, expressed as a number between 0 (impossible) and 1 (certain).

Key idea: The meaning of 'probability' has changed over time from judging trustworthiness to measuring evidence.

The word "probability" comes from the Latin word "probabilitas." Long ago, this word was linked to "probity," which meant how trustworthy someone was, especially in legal cases. It was often connected to a person's social standing.

This is quite different from how we use "probability" today. Now, it is about how much evidence we have for something. We use logical thinking and observations to figure out how likely something is, rather than judging a person's character.

Key idea: Probability can be understood theoretically by counting outcomes, or practically through objective (frequency based) or subjective (belief based) interpretations.

When we talk about how likely something is, we can think about it in different ways. One way is theoretical probability. This is when we can clearly list all possible outcomes and figure out the chances beforehand. For instance, if you roll a standard six sided die, there are six possible outcomes (1, 2, 3, 4, 5, 6). The probability of rolling a 3 is 1 out of 6, or 1/6.

Let us say you flip a coin twice. The possible outcomes are: heads then heads (HH), heads then tails (HT), tails then heads (TH), or tails then tails (TT). There are 4 total outcomes. The probability of getting heads then heads (HH) is 1 out of 4, or 0.25 (25%). The probability of getting at least one head (HH, HT, TH) is 3 out of 4, or 0.75 (75%).

However, in real life, things are not always so clear. There are two main ways people think about probability when it comes to practical situations: objective probability and subjective probability.

Objective probability often looks at how often an event happens over many, many trials. For example, if you flip a coin a thousand times, you would expect heads to come up about half the time. This is called frequentist probability. Another objective view is propensity probability, which sees probability as a natural tendency of an experiment to produce a certain result, even if you only do it once.

Subjective probability is about how strongly someone believes an event will happen. It is a personal degree of belief. For example, a weather forecaster might say there is a 70% chance of rain based on their expert judgment and available data. This is often used in Bayesian probability, where new information updates an initial belief.

“In real life, there are two main ways people think about probability: objective probability and subjective probability.”

Key idea: The mathematical study of probability began with early interest in games of chance and was formalized by mathematicians in the 1600s and beyond.

The mathematical study of probability is actually quite new. People have gambled for centuries, showing an early interest in chance, but it took a long time for mathematicians to create exact ways to describe it.

One important figure was Gerolamo Cardano in the 1500s. He figured out how to define odds by comparing favorable outcomes to unfavorable ones. This led to the idea that the probability of an event is the number of good outcomes divided by the total number of possible outcomes.

Later, in the 1600s, mathematicians like Pierre de Fermat and Blaise Pascal started exchanging letters about games of chance. Their discussions laid the groundwork for the modern theory of probability. Over the next centuries, many other brilliant minds, including Jakob Bernoulli, Abraham de Moivre, and Pierre-Simon Laplace, further developed this field, making it a solid branch of mathematics.

Key idea: Probability is widely applied in real life for risk assessment, financial decisions, scientific research, product reliability, and even language technology.

Probability theory is not just for math class; it is used in many parts of our daily lives. It helps us understand and manage risks.

For example, insurance companies use probability to figure out how likely you are to get into a car accident or get sick. This helps them decide how much to charge for insurance. In finance, traders use probability to guess how likely certain events (like political conflicts) are to affect stock prices.

Governments use probability for things like environmental rules and planning for future needs. Biologists use it to study how diseases spread or how traits are passed down in genetics. Even the games in casinos are designed using probability to ensure the house always has an advantage, while still giving players enough wins to keep them interested.

Another important use is in making products reliable. When you buy a car or a phone, engineers use probability to design it so it is less likely to break. This also helps companies decide how long their product warranties should be.

Even the technology that helps your phone predict the next word you type uses probability! These are called statistical language models, and they calculate the likelihood of certain words appearing together.

Key idea: The mathematical treatment of probability defines a sample space of all possible outcomes and assigns probabilities to specific events within that space.

In mathematics, we think of an "experiment" as something that can have different results. The collection of all possible results is called the sample space. For example, if you roll a die, the sample space is {1, 2, 3, 4, 5, 6}.

An event is a specific outcome or a group of outcomes from the sample space. For example, rolling an odd number on a die is an event, which includes the outcomes {1, 3, 5}.

Probability assigns a value between 0 and 1 to each event. The probability of the entire sample space (meaning something will happen) must be 1. Also, if you have events that cannot happen at the same time (like rolling a 1 and rolling a 2 on a single die roll), the probability of either one happening is just the sum of their individual probabilities.

We often write the probability of an event A as P(A).

The opposite of an event A is "not A." For example, if A is rolling a 6, then "not A" is not rolling a 6. The probability of "not A" is 1 minus the probability of A. So, the chance of not rolling a 6 is 1 minus 1/6, which is 5/6.

When two events, A and B, happen together, we call this their joint probability. If these events do not affect each other (they are independent events), then the probability of both happening is found by multiplying their individual probabilities.

“The probability of "not A" is 1 minus the probability of A.”

Key idea: How you combine probabilities depends on whether events can happen at the same time (mutually exclusive) or not.

Let us look at some common ways probabilities combine.

If two events cannot happen at the same time, they are called mutually exclusive events. For example, you cannot roll both a 1 and a 2 on a single die roll. The probability of both of them happening is 0. The probability of either one happening is simply the sum of their individual probabilities.

For example, the chance of rolling a 1 or a 2 on a six sided die is P(1) + P(2) = 1/6 + 1/6 = 2/6, or 1/3.

What if events are not mutually exclusive? Meaning, they can happen at the same time. For example, drawing a heart or a face card (Jack, Queen, King) from a deck of cards. Some cards are both hearts and face cards (the Jack of Hearts, Queen of Hearts, King of Hearts).

To find the probability of getting a heart or a face card, you add the probability of getting a heart to the probability of getting a face card. But then you must subtract the probability of getting a card that is both a heart and a face card, because you counted those cards twice.

In a standard deck of 52 cards: 13 are hearts, 12 are face cards, and 3 are both (Jack, Queen, King of Hearts). So, the probability is 13/52 + 12/52 - 3/52 = 22/52, which simplifies to 11/26.

Key idea: Conditional probability measures the chance of an event happening when we already know another event has occurred.

Conditional probability is about the likelihood of an event happening given that another event has already occurred. We write this as P(A | B), which means "the probability of event A happening, given that event B has already happened."

Imagine you have a bag with 2 red balls and 2 blue balls. The probability of picking a red ball first is 2 out of 4, or 1/2. Now, let us say you picked a red ball and did not put it back. There are now only 3 balls left in the bag: 1 red and 2 blue.

The conditional probability of picking another red ball, given that you already picked one red ball, is now 1 out of 3, or 1/3. The probability changed because the first event (picking a red ball) affected the possibilities for the second event.

If you had picked a blue ball first instead, then there would be 2 red balls and 1 blue ball left. The conditional probability of picking a red ball given that you picked a blue ball first would be 2 out of 3, or 2/3.

“Conditional probability is about the likelihood of an event happening given that another event has already occurred.”

Key idea: Probability helps us understand complex systems and the inherent randomness found in quantum mechanics, even if we cannot predict every detail.

In a world based on old ideas of physics, if we knew everything about how something started, we would know exactly what would happen next. For example, if you knew the exact force and spin when a roulette wheel was spun, you might predict where the ball would land. But in reality, it is almost impossible to know all those tiny details.

This is why probability is so useful. It helps us understand systems that are too complex to predict perfectly, even if they are technically deterministic. Think about the air in a room: it has countless tiny molecules moving around. We cannot track each one, but probability helps us understand the overall behavior of the air.

Probability is also essential in quantum mechanics, which describes the behavior of matter at a very tiny level (sub atomic particles). At this scale, things are inherently random. We cannot predict exactly what a particle will do, only the probability of it doing something. Even Albert Einstein, a brilliant scientist, famously said, "God does not play dice," because he found this idea of inherent randomness hard to accept. But it is a core part of how the universe works at its most fundamental level.