independance / mutually exclusive question

[waxydock]

Given that two events A and B have probabilities .3 and .7.

a) If A and B are independent, find P(A U B) and

b) If A and B are mutually exclusive, find the probability that neither A nor B occurs.

For a) i think you need to use this:
P(A U B) = P(A) + P(B) - P(A and B)
which gives .3 + .7 - 0.21, since P(A and B) = 0.3x0.7.

And for b) all i can think of is:
P(A^c) and P(B^c)
= 1-P(A) and 1-P(B)
=.7 and .3
=? is it 1

Please help, thanks

 

[JakeD]

Given that two events A and B have probabilities .3 and .7.

a) If A and B are independent, find P(A U B) and

b) If A and B are mutually exclusive, find the probability that neither A nor B occurs.

For a) i think you need to use this:
P(A U B) = P(A) + P(B) - P(A and B)
which gives .3 + .7 - 0.21, since P(A and B) = 0.3x0.7.

And for b) all i can think of is:
P(A^c) and P(B^c)
= 1-P(A) and 1-P(B)
=.7 and .3
=? is it 1

Please help, thanks

You're correct on (a).

For (b), use 1 - P(A U B). This follows from De Morgan's Laws: A^c and B^c = (A or B)^c. Then you'll need P(A and B) when A and B are mutually exclusive.

 

[waxydock]

I think you may be a bit confused with my writing, sorry about that, what i meant was
P(A to the power of c) and P(B to the power of c), which really means P(Not A) and P(Not B). Does that make sense? to the power of c is just notation for not .

If im wrong please let me know, and please tell us how to solve it.

Thanks for your help

 

[JakeD]

I think you may be a bit confused with my writing, sorry about that, what i meant was
P(A to the power of c) and P(B to the power of c), which really means P(Not A) and P(Not B). Does that make sense? to the power of c is just notation for not .

If im wrong please let me know, and please tell us how to solve it.

Thanks for your help

I understood. A^c means A complement, i.e., not A.

So P(A^c) = 1 - P(A) and P(not A and not B) = P(A^c and B^c) = P((A U B)^c) = 1 - P(A U B). Now use same formula you used on (a) but P(A and B) is different because A and B are mutually exclusive instead of independent.

See De Morgran's Laws. Something like this has to be in your textbook.

 

[galactus]

I think it's better to use P(A') for complement instead of
P(A^c). That's what I'll use.

If A and B are mutually exclusive(can not occur at the same time), then

P(A and B)=0 because P(A or B)=P(A)+P(B)-0

Let P(E)= P(A or B)=0.7+0.3=1

Complement is given by P(E')=1-P(E); 1-1=0

P(E')=P(A' or B')=0

If two events are mutually exclusive they can not be independent and vice versa.

 

[JakeD]

I think it's better to use P(A') for complement instead of
P(A^c). That's what I'll use.

If A and B are mutually exclusive(can not occur at the same time), then

P(A and B)=0 because P(A or B)=P(A)+P(B)-0

Let P(E)= P(A or B)=0.7+0.3=1

Complement is given by P(E')=1-P(E); 1-1=0

P(E')=P(A' or B')=0

If two events are mutually exclusive they can not be independent and vice versa.

Galactus, you should check out De Morgan's Laws too. If E = A or B, E' = A' and B'. Actually, P(A' or B') = 1 here.