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### Intro to Conditional Probability

Intro to Conditional Probability

Intro to Conditional Probability

### Definition, Formula, Properties & Examples ^{[1]}

Conditional probability is known as the possibility of an event or outcome happening, based on the existence of a previous event or outcome. It is calculated by multiplying the probability of the preceding event by the renewed probability of the succeeding, or conditional, event.

Imagine a student who takes leave from school twice a week, excluding Sunday. If it is known that he will be absent from school on Tuesday then what are the chances that he will also take a leave on Saturday in the same week? It is observed that in problems where the occurrence of one event affects the happening of the following event, these cases of probability are known as conditional probability.

As depicted by the above diagram, sample space is given by S, and there are two events A and B. In a situation where event B has already occurred, then our sample space S naturally gets reduced to B because now the chances of occurrence of an event will lie inside B.

### Conditional Probability ^{[2]}

In this section, we discuss one of the most fundamental concepts in probability theory. Here is the question: as you obtain additional information, how should you update probabilities of events? For example, suppose that in a certain city, $23$ percent of the days are rainy

Now that you have this extra piece of information, how do you update the chance that it rains on that day? In other words, what is the probability that it rains given that it is cloudy? If $C$ is the event that it is cloudy, then we write this as $P(R | C)$, the conditional probability of $R$ given that $C$ has occurred. It is reasonable to assume that in this example, $P(R | C)$ should be larger than the original $P(R)$, which is called the prior probability of $R$

Let $A$ be the event that the outcome is an odd number, i.e., $A=\{1,3,5\}$. Also let $B$ be the event that the outcome is less than or equal to $3$, i.e., $B=\{1,2,3\}$

### Expert Maths Tutoring in the UK ^{[3]}

A intersection B is a set that contains elements that are common in both sets A and B. The symbol used to denote the intersection of sets A and B is ∩, it is written as A∩B and read as ‘A intersection B’

A∩B can be determined easily by checking the elements that are present in both A and B. Let us go through the concept of A∩B with the help of some solved examples for a better understanding.

The formula A intersection B represents the elements that are present both in A and B and is denoted by A∩B. So, using the definition of the intersection of sets, A intersection B formula is:

### Definition, Formula, Properties & Examples ^{[4]}

Conditional probability is known as the possibility of an event or outcome happening, based on the existence of a previous event or outcome. It is calculated by multiplying the probability of the preceding event by the renewed probability of the succeeding, or conditional, event.

Imagine a student who takes leave from school twice a week, excluding Sunday. If it is known that he will be absent from school on Tuesday then what are the chances that he will also take a leave on Saturday in the same week? It is observed that in problems where the occurrence of one event affects the happening of the following event, these cases of probability are known as conditional probability.

As depicted by the above diagram, sample space is given by S, and there are two events A and B. In a situation where event B has already occurred, then our sample space S naturally gets reduced to B because now the chances of occurrence of an event will lie inside B.

### Conditional probability ^{[5]}

In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred.[1] This particular method relies on event B occurring with some sort of relationship with another event A. In this event, the event B can be analyzed by a conditional probability with respect to A

This can also be understood as the fraction of probability B that intersects with A, or the ratio of the probabilities of both events happening to the “given” one happening (how many times A occurs rather than not assuming B has occurred): .[3]. For example, the probability that any given person has a cough on any given day may be only 5%

For example, the conditional probability that someone unwell (sick) is coughing might be 75%, in which case we would have that P(Cough) = 5% and P(Cough|Sick) = 75 %. Although there is a relationship between A and B in this example, such a relationship or dependence between A and B is not necessary, nor do they have to occur simultaneously.

### 2.1.3.2.5 – Conditional Probability ^{[6]}

A conditional probability is the probability of one event occurring given that a second event is known to have occurred. This is communicated using the symbol \(\mid\) which is read as “given.” For example, \(P(A\mid B)\) is read as “Probability of A given B.”

In the examples below, note that we’re only interested in the events in one row or column.. The two-way contingency table below displays Penn State World Campus enrollments from Fall 2019 in terms of academic level (undergraduate and graduate) and state residency (Pennsylvania and non-Pennsylvania).

We know the individual is an undergraduate student, so we will only look at the row containing the 8360 undergraduate students. Of those 8360 undergraduate students, 3757 were Pennsylvania residents.

### 3.5: Conditional Probabilities ^{[7]}

What do you think is the probability that a man is over six feet tall? If you know that both of his parents were tall, would you change your estimate of the probability? If you know that both of his parents were short, would that affect your estimate in a different way? Most likely. The chance a man is over six feet tall is probably higher if he has tall parents and lower if he has short parents

While conditional probability may seem like a difficult concept, we use it all the time in our every day life.. – Weather forecasters use conditional probability to predict the likelihood of future weather conditions given current conditions

– Sports betting companies may use conditional probability to set the odds that particular teams win their game. These odds may rely on knowledge about the team such knowing that a key player is injured.

### Conditional Probability ^{[8]}

In this section, we discuss one of the most fundamental concepts in probability theory. Here is the question: as you obtain additional information, how should you update probabilities of events? For example, suppose that in a certain city, $23$ percent of the days are rainy

Now that you have this extra piece of information, how do you update the chance that it rains on that day? In other words, what is the probability that it rains given that it is cloudy? If $C$ is the event that it is cloudy, then we write this as $P(R | C)$, the conditional probability of $R$ given that $C$ has occurred. It is reasonable to assume that in this example, $P(R | C)$ should be larger than the original $P(R)$, which is called the prior probability of $R$

Let $A$ be the event that the outcome is an odd number, i.e., $A=\{1,3,5\}$. Also let $B$ be the event that the outcome is less than or equal to $3$, i.e., $B=\{1,2,3\}$

### Conditional Probability and Independent Events ^{[9]}

Suppose a fair die has been rolled and you are asked to give the probability that it was a five. There are six equally likely outcomes, so your answer is 1/6

Since there are only three odd numbers that are possible, one of which is five, you would certainly revise your estimate of the likelihood that a five was rolled from 1/6 to 1/3. In general, the revised probability that an event A has occurred, taking into account the additional information that another event B has definitely occurred on this trial of the experiment, is called the conditional probability of A given B and is denoted by The reasoning employed in this example can be generalized to yield the computational formula in the following definition.

of A given B, denoted , is the probability that event A has occurred in a trial of a random experiment for which it is known that event B has definitely occurred. It may be computed by means of the following formula:

### Conditional Probability: Definition, Properties and Examples ^{[10]}

Being a classical concept in probability theory, conditional probability is one of the prominent approaches to measuring the probability of occurrence of an event, provided that another event has occurred.. First, let’s catch a quick introduction to the concept of probability.

When we say that there are “20% chances”, we are quantifying some events and use words like impossible, unlikely, even like, likely, and certain to measure the probability.. Probability is simply the measure of the likelihood that an event will occur

The sum of all probabilities of all the events in a sample space is equal to 1.. For example, the probability of event A is the sum of the probabilities of all the sample points in event A and denoted by P(A).

### 3 Conditional Probability ^{[11]}

Compute the probability that a randomly selected U.S. Compute the probability that a randomly selected U.S

Is it the same as the previous part? Is it one minus the previous part?. Describe in words the probability that results from subtracting the answer to the first part from 1.

\text{P}(A|B) = \frac{\text{P}(A\cap B)}{\text{P}(B)}. The conditional probability \(\text{P}(A|B)\) represents how the likelihood or degree of uncertainty of event \(A\) should be updated to reflect information that event \(B\) has occurred.

### how to calculate the following conditional probability ^{[12]}

There are two events involved, say event A and event B. I want to know the probability of event B conditioned on the event A

Moreover, once A happened, the events A and B are basically independent. For example, once a person is born, the probability of that person’s death only has relation with the specific time when that person is born, not with the event of the birth of that person

“B cannot happen until A happens” is the same as ‘B is contained in A’ or ‘B is a subset of A’ or ‘B is equal to A’. When one event is a subset of another, they cannot be independent.

### Sources

- https://byjus.com/maths/conditional-probability/#:~:text=Conditional%20probability%20is%20known%20as,succeeding%2C%20or%20conditional%2C%20event.
- https://www.probabilitycourse.com/chapter1/1_4_0_conditional_probability.php#:~:text=Below%2C%20we%20formally%20provide%20the,P(B)%3E0.
- https://www.cuemath.com/probability-a-intersection-b-formula/#:~:text=x%20%E2%88%88%20B%7D-,What%20Is%20P(A%E2%88%A9B)%20Formula%3F,and%20%22B%22%20happening%20together.
- https://byjus.com/maths/conditional-probability/
- https://en.wikipedia.org/wiki/Conditional_probability
- https://online.stat.psu.edu/stat200/lesson/2/2.1/2.1.3/2.1.3.2/2.1.3.2.5
- https://math.libretexts.org/Courses/Prince_Georges_Community_College/MAT_1130_Mathematical_Ideas_Mirtova_Jones_(PGCC%3A_Fall_2022)/03%3A_Probability/3.05%3A_Conditional_Probabilities
- https://www.probabilitycourse.com/chapter1/1_4_0_conditional_probability.php
- https://saylordotorg.github.io/text_introductory-statistics/s07-03-conditional-probability-and-in.html
- https://www.analyticssteps.com/blogs/conditional-probability-definition-properties-examples
- https://bookdown.org/kevin_davisross/gsb518-handouts/conditional-probability.html
- https://stats.stackexchange.com/questions/157364/how-to-calculate-the-following-conditional-probability