What is the connection and difference between P B A and P AB ?

p(b|a) is the probability of b occurring under the premise that a occurs, that is, if a is known to have occurred, then the probability of b occurring, and p(ab) is the probability that ab has occurred at the same time, where a may or may not occur.

The key difference between the two is p(b|a) assumes that a occurs is known, but p(ab) does not.

Note p(b|a) is actually calculated as the probability of b occurring, and ab in p(ab) is not necessarily independent of each other, and the independence of events has a special meaning in probability theory.

Here's an example: Suppose a class in grade 6 has half male and half girls, and the boys don't have braids and the girls have braids.

A student X is randomly selected from the class, A means that the student is a girl, and B means that the student is drawn with braids.

p(b|a) indicates the probability that x will have braids if x is known to be a girl, obviously p(b|a)=1, p(ab) means that x is both a girl and has braids, and the occurrence of a is not a prerequisite for calculating the probability, that is, it is possible to draw out a boy, and it is easy to know that p(ab)=1 2.

As for the solution of the problem, we should find p(b|a) Or find p(ab), it is recommended to depend on the specific situation.

The problem lies in the range of probabilities that are discussed, p(b|a) is to discuss the probability of b occurring in the event where a occurs, while p(ab) is the probability that a and b occur at the same time in all events, although the event in question is ultimately discussed that both a and b occur at the same time, but the latter has more cases in the denominator where a does not occur and b occurs.

The difference between the two is in their definition:p(ab) is the probability of ab occurring at the same time, and the probability of ab occurring at the same time is calculated by taking 100% of all events.

p（b|a) is the probability that event b will occur again on the premise that event a has already occurred. The probability of AB occurring at the same time is calculated by 100% of all event A occurrences.

Edge Probability:is the probability that an event will occur, regardless of other events. The edge probability is obtained like this:

In joint probability, those events that are not needed in the final outcome are combined into the full probability of their event and disappear (the full probability is obtained by summing discrete random variables, and the full probability is obtained by integrating continuous random variables). This is called marginalization. The edge probability of a is expressed as p(a) and the edge probability of b is expressed as p(b).

The difference between p(ab) and p(ab) is mainly the concept.

Random events. A B is called the sum event of A and B, which indicates that at least one of the random events A or random events B occurs; Random event A B is called the product event of A and B, which means that random event A and random event B occur at the same time, and usually, we abbreviate A B as AB.

So, p(a b) denotes the probability that at least one of the random event a or random event b will occur, and p(ab) denotes the probability that random event a and random event b occur at the same time.

Note. 1. ab is the probability that a and b occur at the same time, and a and b are the probability that a or b has one or two occurrences.

2. Different expressions: AB is expressed as A B, and A and B are expressed as A B.

3. The calculation formula is different: p(a+b)=p(a b)=p(a)+p(b)-p(ab), p(ab)=p(a)p(b|).a), ab is the probability that a and b occur at the same time, and a and b are the probability that a or b has one or both occurrences.

4. Different expressions: AB is expressed as A B, and A and B are expressed as A B.

5. The calculation formula is different: p(a+b)=p(a b)=p(a)+p(b)-p(ab), p(ab)=p(a)p(b|).a)。

p(ab) denotes the probability that two events will occur together.

p(b|a) is the probability of b occurring under the premise that a occurs, that is, if a is known to have occurred, then b is the probability of b occurring, and p(ab) is the probability of ab occurring at the same time, where a may or may not occur. The key difference between the two is p(b|a) Assumes that a is known to occur, but p(ab) does not, note that p(b|).a) What is calculated is actually the probability of b occurring.

Introduction to Probability: Probability, also known as "probability", is a reflection of random events.

The probability of occurrence. A random event is an event that may or may not occur under the same conditions. For example, from a batch of ** and defective goods, randomly draw a piece, "draw **" is a random event.

Suppose n experiments and observations are carried out on a random phenomenon, in which event a occurs m times, that is, the frequency of its occurrence is m n. After a lot of trial and error, it is common for m n to get closer and closer to a certain constant (this is proved to be detailed in the law of large numbers of Nouri in the town of Bow. This constant is the probability of event a occurring, which is often expressed by p(a).

When p(a) and p(b) are not related, p(ab) = p(a)*p(b); When p(a) and p(b) are related, p(ab) = p(a|b) p(b) or p(ab) = p(b|).a)/p(a)。

Here are some examples:

1) P(a) and P(B) are not related, the probability of student A scoring more than 90 points in the English test is 7 9, and the probability of student B scoring more than 90 points in the English test is 3 7, then the probability of student A and B scoring more than 90 points in the English test at the same time is p(ab)=p(a)*p(b)=(7 9)*(3 Yulu 7)=1 3;

2) P(a) and P(B) are related, student C participates in the long jump competition, the probability of normal performance in the game is 5 6, the probability of abnormal play is 1 6, during training, the long jump distance of student C is normally above the meter, the number of 100 training sessions of the student is counted, the frequency of the long jump distance above the meter is 34, it is known that student C went to participate in a long jump competition at the banquet and played normally, then the probability of his long jump score above the meter is p(ab)=p(a|).b)/p(b)=(34/100)/(5/6)=51/125。