Using Chebyshev's Theorem (and another minor question)

[Hiche]

Homework Statement

If X is a discrete random variable with mean u = 12 and variance = 9, use Chebyshev's Theorem to find an upper bound for P(X = 21).

Homework Equations

The Attempt at a Solution

Now, I'm not sure about this since there are different upper bounds, right?

P(|21 - 3k < 21 < 21 + 3k|) ≤ 1 / k^2. We solve the inner inequality and we get -k < 3 < k. To find an upper bound, do we simply take a value k > 3?

And an unrelated question (just not to make another thread):

Let A and B be two events with P(A) = 3/4 and P(B) = 1/3. Explain why 1/12 ≤ P(A intersection B) ≤ 1/3. How do we approach this exactly?

 

[phiiota]

regarding your second question, just something to think about: what is P(A Union B)? what does this tell you about P(A intersect B)?

 

[Hiche]

P(A U B) = P(A) + P(B) - P(A intersection B) = 3/4 + 1/3 - P(A intersection B) = 13/12 - P(A intersection B). So, when P(A intersection B) = 1/12, P(A U B) = 1 and when it's 1/3, P(A U B) = 3/4. Does it have to do something with that? It looks too obvious but I'm not figuring it out!

 

[phiiota]

the fact that P(A) + P(B) > 1 tells you there has to be some overlap, right? So you're looking at kind of a best case/worst case scenario. How small could the overlap be? How large could the overlap be?