Not to worry. I don't think you are picking on me.

Of course. The question in my mind is whether or not this is the analogy for the goat -BP game.The fact of the matter is, most people simply don't have a gut feeling for probabilities. Probabilities are useful as expressions of long-term trends in outcomes. If you roll a 6-sided die a couple hundred times, about one-sixth of your rolls will be 1, about one-sixth will be 2, and so on.

Mathematically, we say that on your very next roll of that die, you have a 1/6th chance (or probability) of getting, say, a 6. That's thedefinitionof probability.

Yes, of course.

Don't think so. I wouldn't do that with the 6 sided die. I may or may not be making that error in my analysis of this problem, but I don't see the comparison with the die; I see the correct analog as the coin toss. I think I know what you are driving at though.You, on the other hand, like a lot of people, look at it from the very direct point of view that either you get a 6 or you don't.

I'll admit to the possibility that I am viewing this puzzle falsely as one involving dichotomous choices, and I'll admit that I might not be sharp enough to see that it isn't about independent dichotomous events at all, even after you guys have pointed that issue out. I'm just not quite convinced by the example given by JoininNY because an option seems to me is being omitted, there are 4 options not three as enumerated (I think).Two outcomes. And because most people really don't get probabilities at a gut level, everything tends to look to them like "certainly true", "certainly false", or "50/50"---that is, probability 100%, 0%, or 50%---even though that's not what the math says about the situation.

AGREELet me give another example---you're playing poker and you have four of a kind. You are about to rake in a huge pot against one guy still betting into you. Maybe he has a royal flush and maybe he doesn't. Those are thepossibilities, but it is certainlynota 50% chance that he has a royal flush.

Agree again.In fact, I think the odds of a royal flush are about 650,000:1 against, but if hedoespull that royal flush, those low odds aren't going to be much consolation to you...

Couldn't agree more.By the way (and this may be a rather tangential aside), these are not just simple little game-playing consequences.

Yes, but I think the policy issue really wasn't one of odds. Again, tangentially, you wouldn't want to be either the passenger or the politician involved in the decision if one went down. I think your point is well intended and well made however. These are indeed not trivial problems for fun and games and are best left to the mathematical statisticians when money and lives are at stake.The fact that people in positions of power and authority don't understand odds bites folks in the butt all the time. What were the odds of a plane being downed by the volcanic ash over Europe this past week? Pretty slim; maybe even infinitesimal. But because of the fact that EU functionaries can't tell the difference between "some small risk of disaster" and "near certainty of disaster," the EU bureaucracy made a decision that cost the airline industry a couplebilliondollars.

</math_rant>

Now back to the puzzle-- is there not a choice being left out in Joinin's analysis? Or is what you guys are driving at the business end of "degrees of freedom, n-1"???

I know I'm over my head when I play with mathematicians and statisticians, but I'm not finding the plain English explanations convincing---though if you guys are in fact real world working statisticians I will have no choice but to bow to your professional opinion. I guess you have no obligation to convince a "dunce." Just as Einstein had no obligation to convince anyone outside the world of physics--only the guy calculating yield-- that his unintuitive conclusions were correct.