## How does conditional probability work?

Probability theory is a branch of mathematics that deals with the study of random events and their likelihood of occurrence. Conditional probability is a concept within probability theory that measures the probability of an event occurring given that another event has already occurred.

In other words, conditional probability is the probability of an event happening, taking into account the knowledge that another event has occurred. It is an important concept in statistics, machine learning, and other areas of science where the outcome of one event may depend on the outcome of another.

Conditional probability works by using information about one event to update our knowledge and estimate the probability of another event occurring.

In the case of two events, A and B, we can calculate the probability of A given that B has occurred by dividing the probability of both A and B occurring by the probability of B occurring. This formula allows us to update our prior knowledge about the likelihood of event A, based on the new information we have about event B.

For example, let's say we are interested in the probability of someone having a heart attack, given that they are a smoker. We know from previous studies that the probability of someone having a heart attack is 0.10 (10%), and the probability of someone being a smoker is 0.25 (25%). However, we also know that smokers are at a higher risk of heart attacks, and that the probability of a smoker having a heart attack is 0.25 (25%).

Using conditional probability, we can calculate the probability of someone having a heart attack given that they are a smoker as follows:

P(HA | S) = P(HA ∩ S) / P(S)

where P(HA | S) is the probability of having a heart attack given that the person is a smoker, P(HA ∩ S) is the probability of both having a heart attack and being a smoker, and P(S) is the probability of being a smoker.

Assuming that the probabilities we have are correct, we can substitute in the values we know:

P(HA | S) = 0.25 / 0.25 = 1

This tells us that if someone is a smoker, the probability of them having a heart attack is 1 or 100%. This is a significant increase from the overall probability of 10%.

Conditional probability can also be used to update our knowledge about events that are dependent on each other. For example, let's say we are interested in the probability of flipping two coins and getting at least one head. If we know that the first coin is heads, then the probability of getting at least one head in the second coin flip is now 1/2, rather than the 3/4 probability that would have been the case if we did not have the information about the first coin flip.

In summary, conditional probability allows us to use information about one event to update our knowledge and estimate the probability of another event occurring. It is a powerful tool that can be used in a wide range of fields and applications.

## Example of conditional probability

An example of conditional probability is rolling two dice and finding the probability of getting a sum of 7 given that one of the dice is a 4. The probability of getting a sum of 7 with two dice is 6/36 or 1/6, since there are six possible outcomes that can result in a sum of 7 out of 36 total outcomes. However, if we know that one of the dice is a 4, then we can eliminate any outcomes that do not involve a 4, leaving us with five possible outcomes: (4,3), (4,5), (4,6), (3,4), and (5,4). Out of these five outcomes, only one results in a sum of 7, which is (4,3). Therefore, the conditional probability of getting a sum of 7 given that one of the dice is a 4 is 1/5, or 0.2. This example illustrates how knowledge of one event can affect the probability of another event occurring.

## What is the difference between probability and conditional probability?

The main difference between probability and conditional probability is that probability is the likelihood of an event occurring, without any prior knowledge or assumption, while conditional probability is the likelihood of an event occurring, given that another event has already occurred.

Probability is a fundamental concept in probability theory that refers to the likelihood of an event occurring, expressed as a number between 0 and 1. The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For example, the probability of flipping a coin and getting heads is 1/2, because there is only one favorable outcome (heads) out of two possible outcomes (heads or tails).

Conditional probability, on the other hand, is a concept in probability theory that refers to the probability of an event occurring, given that another event has already occurred. It is calculated by dividing the probability of the joint occurrence of the two events by the probability of the conditioning event. For example, the probability of getting a sum of 7 on two dice, given that one of the dice is a 4, is an example of conditional probability.

In essence, probability deals with the likelihood of an event occurring in general, while conditional probability deals with the likelihood of an event occurring, given some additional information or condition. Probability can be thought of as a special case of conditional probability, where no additional information or condition is given.

To summarize, the main difference between probability and conditional probability is that probability deals with the likelihood of an event occurring in general, while conditional probability deals with the likelihood of an event occurring, given some additional information or condition.

## What is Baye's Theorem?

Bayes' Theorem, also known as Bayes' Rule or Bayes' Law, is a mathematical formula used to calculate the conditional probability of an event, given some prior knowledge or information. The theorem is named after Thomas Bayes, an 18th-century English statistician and philosopher.

Bayes' Theorem states that the probability of event A given event B can be calculated as follows:

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

where P(A|B) is the probability of event A occurring given that event B has occurred, P(B|A) is the probability of event B occurring given that event A has occurred, P(A) is the prior probability of event A occurring, and P(B) is the prior probability of event B occurring.

The formula works by using the prior probabilities and the conditional probabilities of the two events to update our knowledge and estimate the probability of one event given the occurrence of another event.

To illustrate this, let's consider an example. Suppose we have a bag containing 10 red balls and 5 blue balls. We randomly select one ball from the bag, but before we look at its color, we flip a coin. If the coin lands heads, we put the ball back in the bag and add another blue ball. If the coin lands tails, we put the ball back in the bag and add another red ball. We then randomly select another ball from the bag. Given that the second ball we select is blue, what is the probability that the coin landed heads?

Using Bayes' Theorem, we can calculate the probability as follows:

P(heads|blue) = P(blue|heads) * P(heads) / P(blue)

where P(heads|blue) is the probability that the coin landed heads given that the second ball we select is blue, P(blue|heads) is the probability of selecting a blue ball given that the coin landed heads, P(heads) is the prior probability of the coin landing heads, and P(blue) is the prior probability of selecting a blue ball.

From the problem, we know that the probability of selecting a blue ball given that the coin landed heads is 1/3, the prior probability of the coin landing heads is 1/2, and the prior probability of selecting a blue ball is 7/30 (computed by summing the probabilities of selecting a blue ball with heads and a blue ball with tails). We can substitute these values into the formula to obtain:

P(heads|blue) = (1/3 * 1/2) / (7/30) = 5/7

This tells us that the probability of the coin landing heads given that we selected a blue ball is 5/7 or approximately 71%.

Bayes' Theorem is a powerful tool in probability theory and statistics, and is widely used in various fields, such as machine learning, genetics, and medical diagnosis, among others.

## Conditional probability vs. joint probability and marginal probability

Conditional probability, joint probability, and marginal probability are all fundamental concepts in probability theory. While they are related to each other, they represent different types of probabilities and serve different purposes.

### Conditional Probability:

Conditional probability is the probability of an event occurring, given that another event has already occurred. It is calculated by dividing the probability of the joint occurrence of the two events by the probability of the conditioning event. For example, the probability of getting a sum of 7 on two dice, given that one of the dice is a 4, is an example of conditional probability.

### Joint Probability:

Joint probability is the probability of two or more events occurring together. It is the probability of the intersection of the events and is calculated by multiplying the probabilities of each individual event. For example, the probability of rolling a 4 on one die and a 3 on another die is an example of joint probability, which is calculated as (1/6) * (1/6) = 1/36.

### Marginal Probability:

Marginal probability is the probability of a single event occurring, without taking into account any other events. It is calculated by summing or integrating the joint probabilities over all possible values of the other events. For example, the probability of rolling a 4 on a die is an example of marginal probability, which can be calculated as the sum of the joint probabilities of rolling a 4 on one die and each possible value on the other die.

To better understand the relationship between these concepts, consider the following example:

Suppose we have a bag containing 3 red balls and 2 blue balls. We randomly select one ball from the bag and note its color. We then randomly select another ball from the bag without replacing the first ball. We want to calculate the probability of selecting a red ball on the second draw, given that the first ball drawn was blue.

To solve this problem, we can use conditional probability, joint probability, and marginal probability as follows:

Conditional probability:

The probability of selecting a red ball on the second draw, given that the first ball drawn was blue, can be calculated using conditional probability as follows:

P(red|blue) = P(blue and red) / P(blue)

P(blue and red) is the joint probability of selecting a blue ball followed by a red ball, which is equal to (2/5) * (3/4) = 3/10.

P(blue) is the marginal probability of selecting a blue ball, which is equal to 2/5.

Therefore, P(red|blue) = (3/10) / (2/5) = 3/4 or 0.75.

Joint probability:

The joint probability of selecting a blue ball followed by a red ball can be calculated as follows:

P(blue and red) = P(blue) * P(red|blue)

P(blue) is the marginal probability of selecting a blue ball, which is equal to 2/5.

P(red|blue) is the conditional probability of selecting a red ball given that the first ball drawn was blue, which is equal to 3/4.

Therefore, P(blue and red) = (2/5) * (3/4) = 3/10.

Marginal probability:

The marginal probability of selecting a red ball can be calculated by summing the joint probabilities of selecting a red ball and each possible value of the first ball drawn:

P(red) = P(red and blue) + P(red and red)

= P(blue) * P(red|blue) + P(red) * P(red|red)

= (2/5) * (3/4) + (3/5) * (2/3)

= 3/5

## Conclusion

Conditional probability is a powerful tool for modeling and analyzing complex systems and real-world problems. By considering the probability of an event occurring, given that another event has already occurred, conditional probability allows us to incorporate relevant information and dependencies between events. It is a fundamental concept in probability theory and has numerous applications in fields such as statistics, engineering, and computer science. Through understanding conditional probability, we can make informed decisions based on data and develop predictive models that can be used to solve real-world problems. Overall, the study of conditional probability is essential for anyone working with probability and statistics, and it provides a solid foundation for more advanced topics in these fields.