# Conditional probability equation, how is it derived?

I have to admit I'm struck odd by the this definition:

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

I know conditional probability is the "chance of event a dependant even B happening, given A happens". But really, I don't quite get it... what is meant?

Homework Helper
I have to admit I'm struck odd by the this definition:

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

I know conditional probability is the "chance of event a dependant even B happening, given A happens". But really, I don't quite get it... what is meant?
A simple example: imagine you're throwing a single dice. What is the probability that you'll get an even number under the condition that the number is less than 4?

Well the probability of getting an even number is 3/6 and the probability of getting a number under 4 is 3/6 as well. The final answer is 1/6... but I don't see how this connects to the formula.

cristo
Staff Emeritus
Well the probability of getting an even number is 3/6 and the probability of getting a number under 4 is 3/6 as well. The final answer is 1/6... but I don't see how this connects to the formula.
The answer is not 1/6.Think about the question "what is the probability that you roll an even number under the condition that the number rolled is less than 4"

The given condition is that the number rolled is less than 4, ie it is 1,2 or 3. Now, you want the probability that the number rolled is even. Since there is only on even number in the set {1,2,3} then the answer to the given question is 1/3.

This has everything to do with the formula above. Use your values in the equation:

P(even|<4)= P(even and <4)/P(<4) = (1/6)/(3/6)=1/3

edit: sorry to jump in radou.. i didnt see you were still online!

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arildno
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Gold Member
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I have to admit I'm struck odd by the this definition:

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

I know conditional probability is the "chance of event a dependant even B happening, given A happens". But really, I don't quite get it... what is meant?
The crucial point about "conditional probability" is that you move from calculations of probabilities within one set of known information to do calculations within ANOTHER set of known information, and that moving from one knowledge set to another may influence the computed probability.

For example, the probabilities P(B) and P(A and B) are calculated with respect to a set of known information that does NOT include any knowledge of whether A or B has happened.

The probability P(A|B), however, is simply the probability of A happening WITHIN THE INFORMATION SET WHERE THE KNOWLEDGE OF B HAVING HAPPENED is included! (Thus, for example, P(B|B) must necessarily equal 1, and P(not B|B) is necessarily 0)

The relation P(A and B)=P(A|B)*P(B) therefore relates probabilities calculated with respect to different sets of knowledge.

Events of such a nature that the knowledge of either one of them does not influence the probability of the other one occurring, relative to the information set where neither outcome is known, are called INDEPENDENT events.

Ok thanks allot guys! I'm pretty sure I understand now. So the definition of P(A|B) "probability of an element of set A happening in set B". Say, A has 4 elements that are common to set B and set B has 8 elements, then the probability is 4/8. Right?

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arildno