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# Probability question watch

1. Hi, I'm quite confused on the formulas for finding different probabilities. So i used to think that p(A n B) = p(A) x p(B), and that p(AuB)=p(A)+p(B), but then now in S1 i keep coming upon the formula p(AnB) = p(A n B) = P(A) + p(B) - p(a u b). Im really confused on which one to use, and how do you know which to use?

For example, in the question: A and B are two events and p(A)=0.5 and the P(b)=0.2 and p(A n B) = 0.1, find p(A u B). So surely the probability of a and b is 0.5 multiplied by 0.2 which is 0.1 as the question suggests, but then if you do 0.5 add 0.2 for the question which = 0.7, this answer is wrong and you're apparently meant to use another formula? How do you know when and which to use?

Many thanks
2. This relates to the independence or mutual exclusivity of events. The general formula for P(A n B) is P(A n B) = P(A)*P(B|A). However, if the two events are independent, the probability of B happening is not affected by whether A has already happened. Therefore, P(B|A) = P(B) in this case so P(A n B) = P(A)*P(B) when the events are independent.

The second formula P(A u B) = P(A) + P(B) - P(A n B) is the general formula. However, if A and B are mutually exclusive, this means that A and B can’t happen so P(A n B) = 0 and therefore P(A u B) = P(A) + P(B).

Let me know if there’s anything else you want me to explain or try to explain differently.
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3. (Original post by Pepperpeople)
This relates to the independence or mutual exclusivity of events. The general formula for P(A n B) is P(A n B) = P(A)*P(B|A). However, if the two events are independent, the probability of B happening is not affected by whether A has already happened. Therefore, P(B|A) = P(B) in this case so P(A n B) = P(A)*P(B) when the events are independent.

The second formula P(A u B) = P(A) + P(B) - P(A n B) is the general formula. However, if A and B are mutually exclusive, this means that A and B can’t happen so P(A n B) = 0 and therefore P(A u B) = P(A) + P(B).

Let me know if there’s anything else you want me to explain or try to explain differently.
Thanks for the reply. In my example it didn't state whether the results were mutually exclusive or independent so how would you work out the answer to this part? Many thanks
4. (Original post by Bertybassett)
Thanks for the reply. In my example it didn't state whether the results were mutually exclusive or independent so how would you work out the answer to this part? Many thanks
In this case, you know the events are not mutually exclusive since P(A n B) = 0.1. Then you can just rearrange and apply the formula p(A n B) = P(A) + p(B) - p(A u B). If you needed to check two events are independent, you can check if P(B|A) = P(B) since this shows that A has no effect on the probability of B happening. You could use the formula P(A n B) = P(A)*P(B|A) to do this but this isn't needed in this question. Hope this helps
5. (Original post by Pepperpeople)
In this case, you know the events are not mutually exclusive since P(A n B) = 0.1. Then you can just rearrange and apply the formula p(A n B) = P(A) + p(B) - p(A u B). If you needed to check two events are independent, you can check if P(B|A) = P(B) since this shows that A has no effect on the probability of B happening. You could use the formula P(A n B) = P(A)*P(B|A) to do this but this isn't needed in this question. Hope this helps
thanks for the reply. But what if/would it be possible to have an event that isn't mutually exclusive and isn't independant? If so, what would happen in this case? And just to confirm, you only do p(b) + P(a) or use p(b) x p(a) if the event is mutually exlusive? Many thanks.
6. (Original post by Bertybassett)
thanks for the reply. But what if/would it be possible to have an event that isn't mutually exclusive and isn't independant? If so, what would happen in this case? Many thanks.
If two events aren't mutually exclusive or independent then the general formulae P(A n B) = P(A)*P(B|A) and p(A n B) = P(A) + p(B) - p(A u B) are used. Generally you are given enough information in a question to use either of these formulae. These are both in the formula book and independence and mutual exclusivity are special cases of these formulae.
7. (Original post by Pepperpeople)
If two events aren't mutually exclusive or independent then the general formulae P(A n B) = P(A)*P(B|A) and p(A n B) = P(A) + p(B) - p(A u B) are used. Generally you are given enough information in a question to use either of these formulae. These are both in the formula book and independence and mutual exclusivity are special cases of these formulae.
Ok thanks.
And just to confirm, you only do p(b) + P(a) or use p(b) x p(a) if the event is mutually exlusive? Many thanks.
8. (Original post by Bertybassett)
Ok thanks.
And just to confirm, you only do p(b) + P(a) or use p(b) x p(a) if the event is mutually exlusive? Many thanks.
You use p(A u B) = P(A) + p(B) only when the events are mutually exclusive since P(A n B) = 0

You use P(A n B) = P(A)*P(B) only when the events are independent since P(B|A) = P(B)
9. (Original post by Pepperpeople)
You use p(A u B) = P(A) + p(B) only when the events are mutually exclusive since P(A n B) = 0

You use P(A n B) = P(A)*P(B) only when the events are independent since P(B|A) = P(B)
what about for p(A n B) = P(A) + p(B) - p(a u b) and
p(A u B) = P(A) + p(B) - p(a n b)?
10. (Original post by Bertybassett)
what about for p(A n B) = P(A) + p(B) - p(a u b) and
p(A u B) = P(A) + p(B) - p(a n b)?
They are just the same formula rearranged. For mutually exclusive events P(A n B) = 0 so both will end up saying P(A u B) = P(A) + P(B) in this case after rearranging
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11. (Original post by Pepperpeople)
They are just the same formula rearranged. For mutually exclusive events P(A n B) = 0 so both will end up saying P(A u B) = P(A) + P(B) in this case after rearranging
im not sure what you mean by this
12. (Original post by Bertybassett)
im not sure what you mean by this
If you take the formula P(A u B) = P(A) + P(B) - P(A n B), you can rearrange this to get P(A n B) = P(A) + P(B) - P(A u B) by adding P(A n B) to both sides and taking away P(A u B) from both sides. These are two forms of the same formula which you can use to either find P(A n B) or P(A u B) as you need. With mutual exclusivity, P(A n B) = 0 which you can sub into either of the two forms to show that P(A u B) = P(A) + P(B)
13. (Original post by Pepperpeople)
If you take the formula P(A u B) = P(A) + P(B) - P(A n B), you can rearrange this to get P(A n B) = P(A) + P(B) - P(A u B) by adding P(A n B) to both sides and taking away P(A u B) from both sides. These are two forms of the same formula which you can use to either find P(A n B) or P(A u B) as you need. With mutual exclusivity, P(A n B) = 0 which you can sub into either of the two forms to show that P(A u B) = P(A) + P(B)
thanks, so can you always use the formula P(A u B) = P(A) + P(B) - P(A n B) and if an event is mutually exclusive, you can always just sub in zero for p(a n b)?
14. (Original post by Bertybassett)
thanks, so can you always use the formula P(A u B) = P(A) + P(B) - P(A n B) and if an event is mutually exclusive, you can always just sub in zero for p(a n b)?
Yes you can. The general formula P(A u B) = P(A) + P(B) - P(A n B) is in the formula book so you don't need to memorise it. You only need to remember that mutual exclusivity means P(A n B) = 0 and independence means that P(B|A) = P(B)

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