What Is Conditional Probability?

Conditional probability refers to the probability of occurrence of event A under the condition that another event B has occurred. The conditional probability is expressed as: P (A | B), read as "the probability of A under the condition of B". Conditional probabilities can be calculated using decision trees. The fallacy of conditional probability is that P (A | B) is approximately equal to P (B | A). Mathematician John Allen Paulos pointed out in his book Math Blindness that doctors, lawyers, and other well-educated non-statisticians often make such mistakes. This error can be avoided by describing the data with real numbers instead of probabilities.

Conditional Probability

Conditional probability is

Theorem 1

Let A and B be two events, and A is not an impossible event, then

It is the conditional probability that event B occurs under the condition that event A occurs. normally,

, And it meets the following three conditions:

(1) Non-negative; (2) Normative; (3) Additivity.

Theorem 2

Let E be a random test, be the sample space, and A and B be any two events. Let P (A)> 0, say

Is the conditional probability of event B under the condition that "event A occurs".

The above multiplication formula can be generalized to the case of an arbitrary finite number of events.

Assume

,

, ...

For any n events (n2) and

,then

Theorem 3 (

If and only if A and B satisfy

P (AB) = 0

And P (A) 0, P (B) 0

When A and B are

If the probability of event B, P (B)> 0

Then Q (A) = P (A | B) The function Q defined on all events A is

The fallacy of conditional probability is to assume that P ( A | B ) is approximately equal to P ( B | A ). Mathematician John Allen Paulos pointed out in his book Math Blindness that doctors, lawyers, and other well-educated non-statisticians often make such mistakes. This error can be avoided by describing the data with real numbers instead of probabilities.

The relationship between P ( A | B ) and P ( B | A ) is as follows:

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

Here is a fictitious but realistic example. The gap between P ( A | B ) and P ( B | A ) may be surprising and quite obvious.

If we want to tell if an individual has a major illness for early treatment, we may test a large group of people. Although the benefits are clearly visible, at the same time, there is a controversy in the testing behavior, that is, the possibility of detecting false positive results: if there is a person who does not have the disease, but is misdiagnosed as having the disease during the initial test, he may He will be upset and distressed until more detailed tests show that he is not ill. And even after being told that he is actually a healthy person, it may have a negative impact on his life as a result.

The importance of this problem is best explained by the view of conditional probability.

Suppose 1% of the population suffers from the disease, while the others are healthy. We randomly select any individual, and express disease as well as health as well:

P (disease) = 1% = 0.01 and P (well) = 99% = 0.99. Assume that when the test is performed on a person who is not sick, there is a 1% chance that the result will be a false positive (positive is expressed as positive). Meaning:

P (positive | well) = 1%, and P (negative | well) = 99%. Finally, suppose that when the test is performed on a sick person, there is a 1% chance that the result will be false negative (negative means negative ). Meaning:

P (negative | disease) = 1% and P (positive | disease) = 99%. Now, from the calculation: P (negative | well) = 99%

It is the ratio of healthy and negative people in the whole group.

P (positive | disease) = 99% is the proportion of people in the group who are sick and determined to be positive.

Is the rate of false positives among the entire group.

Is the rate of false negatives among the entire group.

Further draws:

Is the rate of positives among the entire group.

P (disease | positive) = 50% is the chance of someone actually getting sick when they are tested positive.

In this example, we can easily see the difference between P (positive | disease) = 99% and P (disease | positive) = 50%: the former is the conditional probability of you being sick and being detected as positive; the latter It's your chance of being tested positive, and you're actually sick. The final result may be unacceptable by the numbers we choose in this case: half of those who are tested positive are actually false positives. ^[4]