Conditional probability equation, how is it derived?

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Discussion Overview

The discussion revolves around the derivation and understanding of the conditional probability equation, specifically P(A|B) = P(A∩B)/P(B). Participants explore the meaning of conditional probability, provide examples, and clarify the relationship between different probability calculations.

Discussion Character

  • Conceptual clarification
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants express confusion about the definition of conditional probability and its implications.
  • One participant provides an example involving a die roll to illustrate the calculation of conditional probability, questioning how it connects to the formula.
  • Another participant corrects the initial misunderstanding about the probability of rolling an even number given the condition of rolling a number less than 4, explaining the correct calculation as 1/3.
  • A later reply emphasizes the importance of understanding that conditional probability involves moving from one set of known information to another, affecting the computed probability.
  • One participant attempts to summarize their understanding of conditional probability using a set-based approach, suggesting a probability calculation based on common elements in sets.

Areas of Agreement / Disagreement

Participants generally agree on the definition and calculation of conditional probability, though initial misunderstandings and corrections indicate some disagreement on specific examples and interpretations.

Contextual Notes

Some participants' calculations and interpretations depend on their understanding of the definitions and the context of the examples provided, which may not be universally agreed upon.

Who May Find This Useful

This discussion may be useful for individuals seeking to understand conditional probability, its derivation, and its application in various examples, particularly in a mathematical context.

Werg22
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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 dependent even B happening, given A happens". But really, I don't quite get it... what is meant?
 
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Werg22 said:
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 dependent 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.
 
Werg22 said:
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!
 
Last edited:
Werg22 said:
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 dependent 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?
 
Last edited:
That is correct.
 

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