Differences between the two concepts of event independence and event incompatibility

This is really a cold, followed by a piece of information on the Internet, so as not to forget again next time.

To truly solve this problem, we must first remember their definitions.

What events are independent?

Event A and B independence means that the probability between the two events satisfies an equation: P (AB) = P (a) P (B)

Event A and B are incompatible, which means that the calculation between the two events satisfies an equation: AB = empty set.

In other words, these two concepts are defined from different perspectives. Independence is from the perspective of probability, and incompatibility is from the relationship Calculation of events.

In addition, there is another point in understanding these two concepts.

If "event a, B independent" is the Chinese description of an object, then "P (AB) = P (a) P (B)" is described in a mathematical language.

Similarly, "event A and B are incompatible" is equivalent to the mathematical language description "AB = empty set"

In terms of the two descriptions, we need to see the Chinese description and reflect the mathematical description. When you see the mathematical description, you must immediately think of the Chinese description.

The above are the differences between the two concepts

Let's take a look at the two contacts.

As we mentioned in the definition

Event A and B are independent, that is, they satisfy "P (AB) = P (a) P (B )"

Event A and B are incompatible, that is, the calculation between two events satisfies an equation: AB = empty set.

Now we can see that two events are independent, does it mean that the events are incompatible?

Based on the incompatibility of events, we obtain the probability of "AB = empty set" on both sides of this equation. We have P (AB) = P (empty set) = 0;

Therefore, if two events are independent of each other, we have P (AB) = P (a) P (B) = P (empty set) = 0

That is, P (a) P (B) = 0 must be met.

Therefore, when P (a) P (B) = 0, A and B are independent of each other to launch a and B.

If the two events are incompatible with each other, then P (AB) = 0 = P (a) P (B), that is, P (a) P (B) = 0

Therefore, when P (a) P (B) = 0, A and B are incompatible before A and B are independent.

In summary, we know that, in general, two incompatible events are not necessarily mutually independent, and they are not necessarily mutually incompatible.

Only when P (a) P (B) = 0 Can the two be introduced to each other.