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How to Determine wether A and B are Mutually Exclusive and Independent

I've been given the information:
P(A) = 0.25 P(B) = 0.4 and P(A'nB') = 0.45

With this information how do I determine if A and B are Mutually Exclusive and/or Independent?
Original post by AdamChris
I've been given the information:
P(A) = 0.25 P(B) = 0.4 and P(A'nB':wink: = 0.45

With this information how do I determine if A and B are Mutually Exclusive and/or Independent?


Use the Event Independence Test.

Two events are independent if, and only if, P(A∣B)=P(A)P(A\lvert B)=P(A), and P(A∩B)=P(A)P(B)P(A\cap B)=P(A)P(B) also follows from it.

P(A∣B)=P(A∩B)P(B)\displaystyle P(A\lvert B) = \frac{P(A\cap B)}{P(B)}
(edited 7 years ago)
Reply 2
Avatar for AdamChris
AdamChris
OP
Original post by RDKGames
Use the Event Independence Test.

Two events are independent if, and only if, P(A∣B)=P(A)P(A\lvert B)=P(A), and P(A∩B)=P(A)P(B)P(A\cap B)=P(A)P(B) also follows from it.

P(A∣B)=P(A∩B)P(B)\displaystyle P(A\lvert B) = \frac{P(A\cap B)}{P(B)}


Is P(AnB) the same as 1- P(A'nB')?
Original post by AdamChris
Is P(AnB) the same as 1- P(A'nB':wink:?


Yes. I think. Not the best at probability so I'm unsure.

Also if two events are mutually exclusive then P(A∩B)=0P(A \cap B) = 0 (ie: the probability of A and B happening at the same time is 0)
(edited 7 years ago)
Reply 4
Avatar for B_9710
B_9710
18
Original post by AdamChris
Is P(AnB) the same as 1- P(A'nB':wink:?


No. A′∩B′=(A∪B)′ A'\cap B' = (A\cup B)' .
You need to find P(A∩B) P(A\cap B ) to see the events are independent.
(edited 7 years ago)
Reply 5
Avatar for Zacken
Zacken
22
As above, note that P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B) So P(A∪B)=0.65−P(A∩B)P(A \cup B) = 0.65 - P(A \cap B)

and P(A′∩B′)=1−P(A∪B)⇒P(A' \cap B') = 1 - P(A \cup B) \Rightarrow can you take it from there?
Reply 6
Avatar for simon0
simon0
13
In addition to what others have stated above, note De Morgan's Law:

A′∩B′=(A∪B)′. A' \cap B' = (A \cup B)'.

(Source: http://www.math-only-math.com/proof-of-de-morgans-law.html )

Find P(A∩B) P(A \cap B) and use the formula for independence of two events and what it means if two events are mutally exclusive to each other.
(edited 7 years ago)

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