How to prove events are independent? S1

If the events are independent then:

$P(A and B)=P(A).P(B)$

Test for independence is:
if P(A <intersection> B) = P(A) P(B)
or P(B|A)= P(B)
they are independent.
I suggest you put the value into a Venn diagram to work out P(A intersection B); then use the first condition.

You can make a Venn diagram with the data given
P(B) = 1 - P(B')
The Venn will quickly show you P (A n B)

edit: will quickly show you P (A' n B); this is the part of B that is not shared with A...
Then apply the adapted rule :
events are mutually exclusive if P (A' n B)= 0
As this means the chance of them happening at the same time is zero, this is the very definition of mutually exclusive!

Describe to me what your venn looks like... My Venn does prove that they are mutually exclusive ( P(A' n B)= 0 )

OK, P (A) value look good, but you have disregarded the P (B') = 0.7.

P(A n B') + outside = P(B') = 0.7
or
P (all in A but not in B) + all on outside = those not in B = 0.7

So, ignore your P (A' n B) and your "outside" and recalculate the "outside", bearing in mind the above.
Then find P (A' n B)....

Sorry if that is waffle-y!

Okay, I'm sorry to keep asking questions, it may seem like I'm trying to get you to do my homework, but I'm really not, and I genuinely don't understand!

I've got a table with two columns, one for "saloon" and another for "hatchback", and then three rows, "silver, "black", and "other". It's asked me to work out, the probability of a silver hatchback, a hatchback, and a hatchback given that it is silver. I've done all that, but then it asks me to "show that the type of car is not independent of its colour.

I have no idea how to work it out!

Okay, I'm sorry to keep asking questions, it may seem like I'm trying to get you to do my homework, but I'm really not, and I genuinely don't understand!

I've got a table with two columns, one for "saloon" and another for "hatchback", and then three rows, "silver, "black", and "other". It's asked me to work out, the probability of a silver hatchback, a hatchback, and a hatchback given that it is silver. I've done all that, but then it asks me to "show that the type of car is not independent of its colour.

I have no idea how to work it out!

Hi,

I was wondering if someone can explain how I can prove that two events are independent.

I'm given the information:
P(A)= 0.7, P(B) = 0.4, P(A u B) = 0.82


Thank you

They are independent, proof:
A and B independent
<=====> (by definition)
P(A)*P(B) = P(A n B)
<=====> (add 1-P(A)-P(B) to both sides)
1 - P(A) - P(B) + P(A)*P(B) = 1 - P(A) - P(B) + P(A n B)
<=====> (simplify)
(1 - P(A))*(1 - P(B)) = 1 - ( P(A) + P(B) - P(A n B) )
<=====> (simplify as P(A u B) = P(A)+P(B)-P(Anb) )
(1 - P(A))*(1 - P(B)) = 1 - P(A u B)

noticing that the above statement is true as (1-0.7)*(1-0.4) = 0.18 = 1-0.82, thus A B indepenent.

If you can use theories like "A and B are independent is necessary and sufficient to A' and B' are independent", then you can straight away say (1-.07)*(1-.04)=1-.82 thus A' and B' are independent thus A and B are independent.