Question

Probability problem

Answer 1

P(A & B) = 1/2 and P(A' & B')=1/3 and P(A)=p and P(B)=2p then find p ?

Answer 2

options are
1/3
4/9
1/9
7/18

i thought all you have to do is P(A & B)= P(A) X P(B) but its not working :(

Answer 3

is the answer 7/18? that's what i am getting

Answer 4

you can only apply that formula when A and B are independent events, which, unfortunately is not mentioned in this case, so that will not work

Answer 5

hey arnab...how you are getting 7/18?

Answer 6

\[P(A' \cap B')= P( A \cup B)'= 1- P( A \cup B)\]
\[P(A \cup B)= P(A)+P(B)-P(A \cap B)\]
given that P(A)=p, P(B)= 2p, P(AB)=1/2 and P(A'B')=1/3
from that equation 1, we get
\[1-P(A \cup B)= 1/3, P(A \cup B)= 2/3\]
so,\[2/3=p+2p-1/2\]
solve for p

Answer 7

you will get 3p=7/6, so, p=7/18

Answer 8

ohh yeah that seems perfectly right but i am wondering that why is this solution not valid:
2p + p =1, 3p=1, p=1/3
i know thats stupid but seems legit

Answer 9

see, when you are equating 2p+p=1, that means only events A and B are possible.. unfortunately that is not mentioned here, there may be event D,E,F etc that have some probability of occurrence too, so p+2p=1 may not be the case all the time

Answer 10

got it?

Answer 11

thanks aranb but i am having trouble finding the proof of the first line in your solution...

Answer 12

@Arnab09

Answer 13

you know the identity right?
\[A' \cap B'= (A \cup B)'\]

Answer 14

naa i dont know that....if you can give me a pointer where i can see the proof......because thats also important

Answer 15

you want rigorous proof right?

Answer 16

yes...i want to learn that

Answer 17

i don't know link, but i can give you the proof. do you know this?
\[if A \subseteq B and B \subseteq A\]
then A=B

Answer 18

yes

Answer 19

wait while i solve this and upload

Answer 20

ok

Answer 21

I sort of understand this concept using the venn diagrams....but i cant get the rigorous proof as you said...

Answer 22

yes, wait, i am uploading

Answer 23

see that^^

Answer 24

thanks that was very helpful of you!

Answer 25

you are welcome :)