Conditional Probability definition

Conditional probability definition

In probability theory, conditional probability definition is the probability of an event occurring given that another event has already occurred. It is represented by the notation P(A|B), where A is the event of interest, and B is the event that has already occurred. This can be read as “the probability of A given B.”

The formula for conditional probability is P(A|B) = P(A and B) / P(B), where P(A and B) is the probability of both events A and B occurring together, and P(B) is the probability of event B occurring regardless of whether event A has occurred. The formula can also be written as P(A|B) = P(B|A) * P(A) / P(B)

Conditional probability is important in many areas of statistics and decision-making, as it allows us to take into account information we already know when making predictions about future events.

Conditional probability rules

There are a few key rules to keep in mind when working with conditional probability:

The rule of total probability: If A1, A2, …, An are mutually exclusive and collectively exhaustive events, then for any event B, the probability of B happening is the sum of the probabilities of B happening given that each of the Ai have occurred. Mathematically, P(B) = Σ(i=1 to n) P(Ai)P(B|Ai)
The rule of multiplication: If events A and B are independent, then the probability of both A and B happening is the product of the individual probabilities of A and B happening. Mathematically, P(A and B) = P(A)P(B)
The rule of conditional probability: If events A and B are dependent, then the probability of both A and B happening is the product of the probability of B happening and the conditional probability of A happening given that B has occurred. Mathematically, P(A and B) = P(B)P(A|B)
The Bayes’ theorem: P(A|B) = P(B|A) * P(A) / P(B)
The total probability of A is 1 : P(A) = 1

It’s also important to note that conditional probability is only defined when the denominator (i.e. P(B)) is not zero.

Conditional probability examples

Here are a few examples of how to calculate conditional probability:

Example 1: A bag contains 4 red balls and 6 blue balls. A ball is drawn from the bag, and it is found to be red. What is the probability that the next ball drawn will also be red?

Solution: We can begin by using the formula for conditional probability: P(B|A) = P(A and B) / P(A)

In this case, event A is the first ball being red, and event B is the second ball being red.

The probability of event A happening is P(A) = 4/10, or 0.4. This is because there are 4 red balls out of a total of 10 balls in the bag.

The probability of event B happening, given that event A has occurred, is P(B|A) = P(A and B) / P(A) = 3/9, or 0.33. This is because there are 3 red balls left in the bag after the first draw, and the total number of balls is 9.

So, the probability of drawing a second red ball from the bag, given that the first ball drawn was red, is 0.33 or 33%.

Example 2: A study found that 10% of people have a certain genetic disease. It is also known that a diagnostic test for the disease is 90% accurate, meaning that it correctly identifies 90% of people who have the disease and 90% of people who do not have the disease. A person is tested and the test result is positive. What is the probability that the person actually has the disease?

Solution: We can begin by using the formula for conditional probability: P(A|B) = P(A and B) / P(B)

In this case, event A is the person having the disease, and event B is the test result being positive.

The probability of event A happening is P(A) = 0.1, or 10%.

The probability of event B happening, given that event A has occurred, is P(B|A) = 0.9, or 90%.

The probability of event B happening, regardless of whether event A has occurred, is P(B) = P(A and B) + P(not A and B) = (P(A) * P(B|A)) + (P(not A) * P(B|not A))

And the probability of A given B is P(A|B) = P(A and B) / P(B) = (0.1 * 0.9) / P(B)

Therefore, the probability that the person actually has the disease is 9%

These are just a few examples of how to calculate conditional probability. The key is to be clear about the events you are considering and the information you are given.

conditional probability and independence

Conditional probability is the probability of an event occurring given that another event has already occurred. For example, the probability of it raining tomorrow given that it is cloudy today would be a conditional probability. Independence refers to the condition where the occurrence of one event does not affect the probability of another event occurring. Two events are independent if the probability of one event occurring is not affected by whether the other event occurs or not. In probability notation, if A and B are independent events, then P(A|B) = P(A) and P(B|A) = P(B)

conditional probability examples