Smallest value for P(A)=P(B)=P(C) s.t. P(A and B and C) always exceeds 0.95 What is the smallest value for P(A)=P(B)=P(C) s.t. P(A and B and C) always exceeds 0.95?
I have made some attempts:
P(A and B and C)=P(A)P(B|A)P(C|A and B)>0.95. But how should the conditinal probabilities be handled?
or
P(A and B and C)=P(A)+P(B)+P(C)-P(A and B)-P(A and C)-P(B and C)+P(A or B or C)>0.95. It is unknown whether the events are independent or not e.g. P(A and C) is not necassary zero, so how to proceed in this case?
Thanks!

Consider minimizing $P(A\cap B \cap C) $ subject to $P(A) = P(B) = P(C) = p$.
In the following Venn diagram,

$$\eqalign{P(A \cap B \cap C) &= p_3\cr
         P(A) = p &= p_1 + p_2 + p_3 + p_4\cr
         P(B) = p &= p_2 + p_3 + p_5 + p_6\cr 
         P(C) = p &= p_3 + p_4 + p_6 + p_7\cr
         &p_1 + \ldots + p_8 = 1\cr}$$
Given $p$, minimizing $p_3$ with these constraints and all variables nonnegative is a linear programming problem.  When $p$ is near $1$, the 
optimal solution will have $p_1 = p_5 = p_7 = p_8 = 0$, $p_2 = p_4 = p_6 = 1-p$ and $p_3 = 3 p - 2$.  Thus to have $p_3 = .95$ you need $p = 2.95/3 = .98333\ldots$.
