What is meant by minimum and maximum probability of an event. I came across a questions that asks minimum and maximum probability of three heads in three coins are flipped. We need to find minimum and maximum probability in two cases
- When all three coins are not independent
- All pairs of coins are mutually independent
The probability of head and tail on each individual coin is 0.5. I am more concerned on how to approach this problem rather than its solution. So far, I am unable to find any material on minimum and maximum probability.
Best Answer
Assign probability $1/4$ to each of the outcomes TTT, HHT, THH, THH. (That is, spread the mass uniformly on the subset where there is an even number of heads. One can do this by flipping the first two coins independently, and then using the parity of the first two to determine the output of the 3d flip. At any rate, in this case, $P(HHH)=0$. This strange probability law evidently minimizes $P(HHH)$.
Now assign probability $1/2$ to the outcomes HHH and TTT. Now $P(HHH)=1/2$, which is maximal. (If $P(HHH)>1/2$ we'd make the first coin unfair, giving it probability $>1/2$.)
That's the answer. A general principle here is that the joint probability law is constrained by having certain marginals. The constraints are linear (these probabilities must add up to that) and so the OP's problem boils down to a linear programming problem.