Largest and smallest possible values of a probability question

[PhysicsMathGuy]

Hello,
So I've been trying to solve this problem for a while and I can't get my head around it.
"Suppose we choose a positive integer at random, according to some unknown distribution. Suppose we know P({1,2,3,4,5}) = 0.3, P({4,5,6}) = 0.4 and P({1}) = 0.1"

All I've managed to get so far is that P({2,3,4,5}) = P({1,2,3,4,5}) - P({1}) = 0.2
I'm sure that's not a good upper bound because there's probably a way the use the fact that the union of {1,2,3,4,5} and {4,5,6} is {4,5}.

Thanks in advance for your help!

 

[PhysicsMathGuy]

Just saw that this section is not for this. Sorry, my bad!

 

[mathman]

P({4,5}) ≤ P({2,3,4,5}) =0.2, P({6}) = P({4,5,6}) - P({4,5}) ≥ 0.2. What are you trying to find?

 

[PhysicsMathGuy]

Thanks a lot for your reply! I'm trying to find the largest and smallest possible values of P({2}).

 

[mathman]

Off hand P({2}) is between 0 and 0.2. I suspect this is the best you can do since there is no restriction on P({3}) or P({6}).

 

[PhysicsMathGuy]

Thanks a lot! That's what I thought, but I wasn't sure and thought I was missing something. Thanks a lot!