# Probabilities Questions - What is P(A n B) = P(A) x P(B)? **URL:** \ **Category:** Factual Questions\ **Created:** [November 1, 2005, 4:20am UTC](https://boards.straightdope.com/t/probabilities-questions-what-is-p-a-n-b-p-a-x-p-b/328859 "2005-11-01T04:20:30Z")\ **Posts on this page:** 5\ **Page:** 1
**Author:** ![Crowbar\_of\_Irony\_3](https://avatars.discourse-cdn.com/v4/letter/c/f08c70/32.png) [@Crowbar\_of\_Irony\_3](https://boards.straightdope.com/u/Crowbar_of_Irony_3)\ **Post date:** [November 1, 2005, 4:20am UTC](https://boards.straightdope.com/t/probabilities-questions-what-is-p-a-n-b-p-a-x-p-b/328859/1 "2005-11-01T04:20:30Z")
Let A represents a random event, and B represents another event. I understand that if A and B are mutually exclusive, then P(A n B) or the probabilities of A intersect with B will give 0. What’s confusing is at times the lecture notes I have state that P(A n B) is equal to P(A) x P(B), but other times it gives a value for P(A n B), but it isn’t equal to that of P(A) x P(B). So when can I use P(A n B) = P(A) x P(B) to find the intersection of two events? ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [November 1, 2005, 4:34am UTC](https://boards.straightdope.com/t/probabilities-questions-what-is-p-a-n-b-p-a-x-p-b/328859/2 "2005-11-01T04:34:24Z")
> [@ExtraKun](#): > > Let A represents a random event, and B represents another event. I understand that if A and B are mutually exclusive, then P(A n B) or the probabilities of A intersect with B will give 0. > > What’s confusing is at times the lecture notes I have state that P(A n B) is equal to P(A) x P(B), but other times it gives a value for P(A n B), but it isn’t equal to that of P(A) x P(B). So when can I use P(A n B) = P(A) x P(B) to find the intersection of two events? When A and B are _independent_ events, or in other words when the probability of “A given B” is the same as the probability of A by itself. Unfortunately, if you dig a little into the definition of conditional probability (i.e, what I mean when I say the probability of “A given B”) you’ll find that mathematically the statement P(A n B)=P(A) x P(B) is the definition of “A and B are independent”, not the other way around. But you can get a good grasp of the colloquial meaning of “independent”…namely, that the outcome of B has no effect on the outcome of A and vice versa…and you’ll get the idea. For example, if A=“roll 6 on a die” and B=“draw a spade from a deck of cards”, the events are independent and P(A n B)=P(A) x P(B). But if A=“roll 6 on a die” and B=“roll an even number on that same die”, then they’re not independent and P(A n B) will not equal the product of the individual probabilities. ---
**Author:** ![Crowbar\_of\_Irony\_3](https://avatars.discourse-cdn.com/v4/letter/c/f08c70/32.png) [@Crowbar\_of\_Irony\_3](https://boards.straightdope.com/u/Crowbar_of_Irony_3)\ **Post date:** [November 1, 2005, 4:46am UTC](https://boards.straightdope.com/t/probabilities-questions-what-is-p-a-n-b-p-a-x-p-b/328859/3 "2005-11-01T04:46:38Z")
> [@Orbifold](#): > > When A and B are _independent_ events, or in other words when the probability of “A given B” is the same as the probability of A by itself. > > Unfortunately, if you dig a little into the definition of conditional probability (i.e, what I mean when I say the probability of “A given B”) you’ll find that mathematically the statement P(A n B)=P(A) x P(B) is the definition of “A and B are independent”, not the other way around. But you can get a good grasp of the colloquial meaning of “independent”…namely, that the outcome of B has no effect on the outcome of A and vice versa…and you’ll get the idea. > > For example, if A=“roll 6 on a die” and B=“draw a spade from a deck of cards”, the events are independent and P(A n B)=P(A) x P(B). But if A=“roll 6 on a die” and B=“roll an even number on that same die”, then they’re not independent and P(A n B) will not equal the product of the individual probabilities. So what if P(A) and P(B) are dependent? Is there anyway to find P(A n B) if they A and B are dependent? I do know that if A is a subset of B, then P(A n B) = P(A). What if it is not? Is it possible to work out P(A n B) only with just P(A) and P(B) and if A and B are dependent and A is not a subet of B\> ---
**Author:** ![bitwise](https://avatars.discourse-cdn.com/v4/letter/b/ea5d25/32.png) [@bitwise](https://boards.straightdope.com/u/bitwise)\ **Post date:** [November 1, 2005, 4:52am UTC](https://boards.straightdope.com/t/probabilities-questions-what-is-p-a-n-b-p-a-x-p-b/328859/4 "2005-11-01T04:52:11Z")
> [@ExtraKun](#): > > So what if P(A) and P(B) are dependent? Is there anyway to find P(A n B) if they A and B are dependent? I do know that if A is a subset of B, then P(A n B) = P(A). What if it is not? Is it possible to work out P(A n B) only with just P(A) and P(B) and if A and B are dependent and A is not a subet of B\> There is no general way. But you could to do it if you also knew P(A union B). P(A intersection B) = P(A) + P(B) - P(A union B) ---
**Author:** ![Dr\_Paprika](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr_paprika/32/3042_2.png) [@Dr\_Paprika](https://boards.straightdope.com/u/Dr_Paprika)\ **Post date:** [November 1, 2005, 4:54am UTC](https://boards.straightdope.com/t/probabilities-questions-what-is-p-a-n-b-p-a-x-p-b/328859/5 "2005-11-01T04:54:44Z")
> **[Conditional probability](https://en.wikipedia.org/wiki/Conditional_probability)** > > In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. If the event of interest is A and the event B is known or assumed to have occurred, "the conditional probability of A given B", or "the probability of A under the condition B", is usually written as P(A|B) or occasionally PB(A). This can also be understood as the fraction of probability B t For exam...