Jumplion post=18.73797.809688 said:
SNIP
OH, OH, OH! Okay, now I understand, I remember seeing a problem or two like that (most notably in the movie "21"), now I get why it's 33%.
But the one thing that's nagging me is that, lets say, if the other copper mine struck, erm, copper than the other one still has 50% chance without putting the one that struck copper into effect....right? But unless that fuel rod was in direct explosive radius of the other fuel rod, the one that was defective still wouldn't have any real effect on the other one.....
Okay, maybe I don't get it yet.
SNIP
Nah, you get it. My examples were probably not that good, but in both examples the same principal - that knowing the value of one member of a set affects the probability spread of the other members' values - was the same as in the original problem. I was attempting to show (crudely) how this can be important for people other than engineers or mathematicians. If you randomly selected two mine sites and one struck copper, the chance for the second is that you randomly selected two winners, NOT the chance that any one mine will strike copper. Similarly, if you randomly select two fuel rods and one fails, the probability of the other failing is the probability that you randomly selected two defective fuel rods, NOT the chance of any individual fuel rod being defective. This sort of thing can be very important in engineering, although hopefully a critical component with a 50% failure rate will have more than one back-up. In any case, the probability function in question is the composition of the set you selected, not the probability function of each member's possible values.
A more likely scenario would be a system with five widgets, each with a 5% defective rate. If at least three widgets must be in working order for proper operation and one widget is observed to fail, what is the chance two more widgets
from the five randomly selected for the system will fail? Now imagine the system is a $300,000,000 satellite and a repair mission costs $25,000. Do you schedule a repair mission, or do you wait and hope more widgets don't fail? Variations of this kind of problem are reasonably common, and an incorrect understanding of probability and statistics can sink a company.
My point was that understanding set probability versus sequential probability is important for everyone, but it can be VERY important for some.
The same thing can be expanded to include coin tosses. If you toss two coins, you will have a distribution of possible results analogous to the pup gender question. The result of each coin toss is independent from the other. But the probability of the
set of both coin tosses follows a rigid structure. Knowing one coin's result changes the probability of the other coin's result. As Dirtface said, it's the Monte Hall problem.