Help with my 2 easy but hard probability questions

[sneaky666]

1.
Suppose P([1/n,1/2])<= 1/3 for all n = 1,2,3,...
a) Must we have P((0,1/2]) <= 1/3 ?
b) Must we have P([0,1/2]) <= 1/3 ?

My answer - Please correct me if i am wrong
a)

P([1/n,1/2])<= 1/3 for all n = 1,2,3,...
lim P([0,1/2])
n-> infinity

so then P((0,1/2]) is a subset of P([1/n,1/2]) for all n = 1,2,3,...
because any number in (0,1/2] can lie in [1/n,1/2] provided the correct n value is in -> 1,2,3...

so true
b)

P([1/n,1/2])<= 1/3 for all n = 1,2,3,...
lim P([0,1/2])
n-> infinity

so then P([0,1/2]) does not equal P([1/n,1/2]) for all n = 1,2,3,...
because as n approaches infinity, the limit of 1/n goes to 0, but doesn't touch 0

so false

2.

Suppose P((0,1/2]) = 1/3. Prove that there is some n such that P([1/n,1/2]) > 1/4.

lim P([1/n,1/2]) = P([0,1/2]) > 1/3 > 1/4
n -> infinity

i don't think this is entirely right, i sort of guessed...

 

[vela]

1.
Suppose P([1/n,1/2])<= 1/3 for all n = 1,2,3,...
a) Must we have P((0,1/2]) <= 1/3 ?
b) Must we have P([0,1/2]) <= 1/3 ?

My answer - Please correct me if i am wrong
a)

P([1/n,1/2])<= 1/3 for all n = 1,2,3,...
lim P([0,1/2])
n-> infinity

so then P((0,1/2]) is a subset of P([1/n,1/2]) for all n = 1,2,3,...
because any number in (0,1/2] can lie in [1/n,1/2] provided the correct n value is in -> 1,2,3...

so true

P((0,1/2]) and P([1/n,1/2]) are numbers. One is not a subset of the other. Also, you have the relationship backwards: [1/n,1/2]⊂(0,1/2], not (0,1/2)⊂[1/n,1/2] as you claimed.

b)

P([1/n,1/2])<= 1/3 for all n = 1,2,3,...
lim P([0,1/2])
n-> infinity

so then P([0,1/2]) does not equal P([1/n,1/2]) for all n = 1,2,3,...
because as n approaches infinity, the limit of 1/n goes to 0, but doesn't touch 0

so false

Right.

2.

Suppose P((0,1/2]) = 1/3. Prove that there is some n such that P([1/n,1/2]) > 1/4.

lim P([1/n,1/2]) = P([0,1/2]) > 1/3 > 1/4
n -> infinity

i don't think this is entirely right, i sort of guessed...

It's not correct. For one thing, you don't know that P([0,1/2])>1/3.

What kind of properties about sets and set operations as it pertains to probabilities do you know? (This would have been what you should have written under "relevant equations" in the provided template.)