Here's my question that I'm struggling with.

If I've got 4 chances to to achieve X goal, and in each chance that I have I've got a 65% chance of achieving this goal, what is my overall probablity of achieving this goal? Also, what would it be after each of my 4 chances (i.e., for my 2nd chance, would it be (.65 * (.65*.65)? I've forgotten some basic probablity theories, and would appreciate any help on this.

With the probablities above, would it be worth paying $20K for 4 chances to achieve the goal or $8.5K for only 1 chance to acheive the goal? (at 65% probability)

First we calculate the probability that we will NOT achieve the goal in 4 chances. For one chance this probability is 1-0.65=0.35. So, for 4 chances it will be (Not at the first chance, not at the second, not at the third, and not at the fourth) 0.35*0.35*0.35*0.35= 0.015.

Now, our AVERAGE loss if we pay $20K for 4 chances to achieve the goal is $20,000*0.015=$300.125

Our average loss if we pay $8.5K for 1 chance to achieve the goal is $8,500*0.35=$2975.

We see that it is less risky to pay $20K for 4 chances to achieve the goal.