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I'm taking an intro to statistics class and currently we are covering conditional probability. The chapter explained what it is quite well, but it didn't explain how the formula for it was derived, i.e it didn't explain explain how

$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$

was derived.

I would greatly appreciate if someone could show the reasoning as well as derivation of this formula.

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2 Answers 2

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The formula for conditional probability is actually taken as somewhat of an axiom in probability and isn't really derived. It makes sense if you think of conditioning as reducing the sample space of an experiment, and remember that probabilities are always measured in relation to the total sample space. In particular they're always proportions of the total sample space.

When we reduce the sample space to $B$ then the only way for $A$ to occur is if $B$ does as well, hence the $P(A \cap B)$. And the size of this event with respect to the new sample space is $P(A \cap B) / P(B)$, or the fraction of $B$ taken up by $A \cap B$.

It can also help to imagine a Venn diagram when thinking about conditional probability. Whenever you condition on an event you restrict yourself to being in one of the circles, and all conditional probabilities are measured as fractions of the area of that circle.

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    $\begingroup$ Interesting that it is an axiom and cannot be derived. This should be like proving NP=P. I personally appreicate that this student is thinking on first principles and not accepting non-sense. $\endgroup$
    – user46925
    Commented Feb 29, 2016 at 2:42
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When we derive the formula, it becomes easy to see why it is so.

The probability of A occurring given B can be derived easily by first restating the problem by considering ONLY members of B, such as "probability of a member belonging to B having a special attribute (member of A here)". This is the dividing number of members in B having special attribute by the total number in B.

Now, both numerator and denominator can be divided by ANY positive number and the value will not change.

We take the total number of members in the system of A & B now for that 'ANY' number. Then, this becomes definition and formula for 'conditional probability'

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    $\begingroup$ Are you perhaps assuming all elements of the sample space have equal (and nonzero) probabilities? $\endgroup$
    – whuber ♦
    Commented Nov 10, 2023 at 23:21
  • $\begingroup$ @whuber may I know other case examples in this context? $\endgroup$
    – sudhakar
    Commented Nov 14, 2023 at 4:41
  • $\begingroup$ Because this is a theoretical, textbook question, I do not understand what a "case example" might be. But generally, probabilities are not assumed to be discrete and uniform: that's a very special case that applies only to limited models and to simple random sampling of finite populations. $\endgroup$
    – whuber ♦
    Commented Nov 14, 2023 at 13:55
  • $\begingroup$ May I know what is meant by, by way of example, unequal probabilities, at a theoretical textbook question level dealing with conditional probabilities ? $\endgroup$
    – sudhakar
    Commented Feb 29 at 13:36
  • $\begingroup$ I did not understand the reply so went back to your initial question. $\endgroup$
    – sudhakar
    Commented Feb 29 at 13:52

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