Conditional Probability

Conditional probability is a concept in probability theory that determines the likelihood of an event occurring, provided another event has already occurred. It is an important part of probability and statistics because in real-world situations, we may have some information available that affects the outcome of an event.

Basics of probability

Before diving into conditional probability, let's briefly revisit the definition of probability. Probability is a way of measuring the uncertainty associated with a particular event. It can be calculated as follows:

Probability of an event A, P(A) = (Number of outcomes favourable to A) / (Total number of possible outcomes)

So, if we have a fair six-sided dice, the probability of getting a three can be expressed as:

P(roll 3) = 1/6

This is because it has one favorable outcome (3 coming) and a total of six possible outcomes (1, 2, 3, 4, 5, or 6).

Introduction to conditional probability

Conditional probability comes into play when we know that one event has occurred and we want to know what the probability is that another event will occur. Mathematically, it is expressed as:

P(A|B) = P(A and B) / P(B)

Here, P(A|B) is the probability of event A occurring, given that event B has occurred. P(A and B) is the probability that both events A and B occur, and P(B) is the probability that event B occurs.

    P(A|B) = P(A and B) / P(B) = (1/52) / (13/52) = 1/13
    

Visualization of conditional probability

Imagine two circles within a square. The square represents all possible outcomes, while the circle represents the occurrence of two events A and B. The intersection of the two circles represents the probability of both events occurring.

Consider throwing a six-sided dice. Let the event A be 'rolling an even number' and event B be 'rolling a number greater than 3'. The possible outcomes for A are {2, 4, 6}, and for B are {4, 5, 6}.

In this diagram, the shaded intersection region represents the probability of both A and B occurring, that is, getting a number that is both even and greater than 3, which are {4, 6}. The conditional probability P(A|B) is simply the circle centered on B because B occurred, which includes only the part of A that intersects with B.

Additional text examples

Imagine that in a village 30% of the people have cats and 20% have dogs. 10% of the dog owners also have cats. We want to find out what percentage of cat owners also have dogs.

Let C be the event that a person has a cat, and D be the event that a person has a dog.

- P(C) = 0.30 - P(D) = 0.20 - P(C and D) (dog owners who also own cats) = 0.02 (10% of 0.20)

P(C|D) = P(C and D) / P(D) = 0.02 / 0.20 = 0.10

So, of all people who own dogs, 10% also own cats. Now, if we want to find out the reverse - what percent of cat owners also own dogs:

P(D|C) = P(C and D) / P(C) = 0.02 / 0.30 ≈ 0.067

Therefore, about 6.7% of cat owners also own dogs.

Properties of conditional probability

Conditional probability has several properties that follow from its definition:

1. Multiplication Rule: P(A and B) = P(A|B) * P(B) = P(B|A) * P(A)
2. Commutative Property: P(A|B) is not necessarily the same as P(B|A). They are based on different conditions.
3. Bayesian context: Aspects of conditional probability are widely used in Bayesian statistics, to update the probability of a hypothesis as more evidence becomes available.

Application of conditional probability with Bayes theorem

Bayes' theorem is a fundamental theorem in probability theory and statistics that uses conditional probabilities. It provides a way to update one's beliefs about a hypothesis based on given evidence. The theorem states:

P(A|B) = [P(B|A) * P(A)] / P(B)

This formula gets its power because it allows us to invert conditional probabilities when we know the opposite situation.

    P(t) = P(t|d) * P(d) + P(t|¬d) * P(¬d)
         = 0.99 * 0.01 + 0.01 * 0.99
         = 0.0099 + 0.0099
         = 0.0198
    

    P(D|T) = [P(T|D) * P(D)] / P(T)
           = 0.99 * 0.01 / 0.0198
           = 0.0099 / 0.0198
           ≈ 0.5
    

Conclusion

Conditional probability is a cornerstone in probability theory and statistics. It adjusts the likelihood of events based on known conditions, providing a way to continually update our understanding as more information becomes available. In today's world full of uncertainties, this concept helps us predict and make decisions more concretely in fields ranging from healthcare and finance to artificial intelligence and beyond.

Hopefully this explanation has given you a basic understanding of conditional probability. As with all areas of mathematics, practicing with real-world scenarios will deepen your understanding and increase your confidence in applying these concepts.

Comments