Probability is a topic in the world of mathematics that is deeply connected not only to science but also to our daily lives. When we toss a coin, roll dice, or predict whether it will rain—we are actually using the language of probability. Probability tells us how likely an event is to occur.
Basic Definition of Probability
We commonly denote probability by P(X), where X can be any event. Its validity ranges from 0 to 1.
- If P(X) = 0, the event is impossible.
- If P(X) = 1, then the event is certain.
And if it is between 0 and 1, then it means that the event has that much probability of occurring.
For example, if we have a bag containing 10 red balls and 10 black balls, and we draw one ball from it, the probability of drawing a red ball is:
$$
P(Red) = \frac{10}{20} = \frac{1}{2}
$$
Conditional Probability
Now suppose we have two bags (Bag A and Bag B). We want to know the total probability of drawing a red ball.
To solve this situation, we need to use conditional probability. The formula is:
$$
P(red ball) = P(bag A) \times P(red ball | bag A) + P(bag B) \times P(red ball | bag B)
$$
Here, ” | ” means “given the given condition.” For example, $P(red ball | bag A)$ means “the probability of getting a red ball if bag A is chosen.”
If the probability of choosing both bags is the same, i.e. 1/2, and each bag contains 10 red and 10 black balls, then:
$$
P(red ball) = \frac{1}{2} \times \frac{10}{20} + \frac{1}{2} \times \frac{10}{20} = \frac{1}{2}
$$
Intersection and Set Theory
To understand conditional probability, we need to understand the concept of intersection. When two events A and B occur simultaneously, we denote it by $A \cap B$.
The formula for conditional probability is:
$$
P(A|B) = \frac{P(A \cap B)}{P(B)}
$$
This means what is the probability of event A occurring if event B has already occurred.
Some Important Concepts in Probability Theory
- Sample Space: The set of all possible outcomes of an experiment. For example, {heads, tails} when tossing a coin.
- Event: Any outcome selected from within the sample space. For example, getting heads when tossing a coin.
- Probability: A number between 0 and 1 that indicates the likelihood of an event occurring.
Bayes’ Theorem
Bayes’ Theorem is extremely important in probability theory. It is used when we need to calculate the probability of an event, given some prior information (condition).
The formula is:
$$
P(A | B) = \frac{P(B | A) \times P(A)}{P(B | A) \times P(A) + P(B | \neg A) \times P(\neg A)}
$$
To understand this, consider the same example—suppose we know that the ball is red. Now we need to determine whether the ball came from bag A or bag B.
This type of probability is called Posterior Probability* because it is based on prior information. Whereas when we calculate the unconditional probability, it is called Prior Probability*.
Bayes’ theorem was discovered by Reverend Thomas Bayes, and today this theory is widely used in the fields of machine learning and data science.
Probability Density
So far, we have discussed discrete events, i.e., limited probabilities. But in real life, continuous events often occur, such as temperature, time, length, etc.
In such situations, we use the Probability Density Function (PDF).
If the probability of a real number $x$ being in the interval (x, x+dx) is given, then:
$$
P(x \in [a,b]) = \int_{a}^{b} p(x) \, dx
$$
Here $p(x)$ represents the probability density. There are two basic rules:
- $p(x) \geq 0$
- $\int_{-\infty}^{\infty} p(x) \, dx = 1$
Use of Probability in Machine Learning
Now the question arises: how does all this relate to machine learning?
When building models in machine learning, we need to make predictions. Probability helps us understand how likely a prediction is to be correct.
For example, suppose we have temperature data for a region over several days. If we want to predict how many bottles of water will be needed in that region next week, we can use probability and statistical models to make this prediction.
Conclusion
In this article, we covered the fundamentals of probability theor
- The definition of probability and sample space
- Conditional probability and intersection
- Bayes’ theorem and its importance
- Probability density and continuous events
- And its practical application in machine learning
Probability is a mathematical tool that enables us to make decisions in a world of uncertainty. It is extremely useful not only for mathematicians and scientists but also in everyday life.
