Guys, I'd love some help with this math problem that's been bouncing around in my head for a while.
I've posted something similar to this before, and I never really got an answer that I could understand and made sense. It's something that's been bothering me for years, and it's so counter-intuitive that it drives me crazy. If someone with knowledge of statistics says "give up, just accept it, that's the way it is" - that's fine, I can do that. But if there's something deeper here, I'd love to hear about it.
Ok, so here we go.
This statistics problem takes the form of a game. You have a bag with ten marbles. The marbles can either be white or black, but there has to be ten total, and at least one white and at least one black. The way the game goes is this:
You put a combination of ten marbles into the bag, and draw one out. If it's white, the game is over. If it's black, you draw again, and keep drawing one at a time until you get a white one. That then ends the game.
For example, say there's 9 white marbles and 1 black marble. The chance of pulling a white marble on the first draw (ending the game outright) is 9/10, or 90%. The chance of pulling the black marble and then a white marble (to finish the game) is (1/10)*(9/9) = 10%.
Pretty simple stuff. Let's look at an example where there are 8 white marbles and 2 black marbles.
Chance of pulling white marble straight off: 8/10.
Chance of pulling one black, then a white: (2/10)*(8/9)=0.177
Change of pulling two blacks, then a white: (2/10)*(1/9)*(8/8)=0.022
This makes sense to me, logically. It's harder to draw two blacks then white, than it is to draw one black then white. Once you have one black marble out of the bag, you have to avoid all of those 8 white marbles floating around and find the one black marble. That's hard to do, avoiding all those white marbles.
I won't go through all the math because it's redundant, but this theme plays out as you go through 3 black / 7 white, 4 black / 6 white, 5 black / 5 white, 6 black / 4 white, 7 black / 3 white, and 8 black / 2 white. It becomes harder and harder to get all the black marbles out of the bag (avoiding the white ones, which would end the game) before finally hitting white. Hence the decreasing odds for emptying the bag of black marbles before hitting a white one.
This makes perfectly good sense to me, and it's backed up by the math if you'd care to draw it out, like I have above. It becomes *very* difficult to avoid hitting those white marbles until all the black ones are out.
However, something very strange happens when there are 9 black marbles and 1 white marble. Before we draw out the math, think about it: each time, we have to avoid hitting the white marble which becomes more and more difficult as the game continues. But let's look at the math:
Chance of hitting white straight away: 1/10 = 0.1.
Chance of hitting 1 black, then the white marble: (9/10)*(1/9)= 0.1.
Chance of hitting 2 blacks, then the white marble: (9/10)*(8/9)*(1/8)=0.1
......
......
Chance of hitting 5 blacks, then the white marble:
(9/10)*(8/9)*(7/8)*(6/7)*(5/6)*(1/5). As you can probably see, everything cancels, and we're left with odds of 1/10, or 0.1.
Taking it to the far extreme: chance of hitting all 9 blacks, missing the white marble *every single draw* - which in other starting marble ratios, was very difficult to do - before finally there's only one marble left in the bag, the white one, which is pulled out to end the game.
(9/10)*(8/9)*(7/8)*(6/7)*(5/6)*(4/5)*(3/4)*(2/3)*(1/2)*(1/1) = 1/10, or 0.1.
So you have an equal chance of pulling the one white marble out of ten in the bag straight off (10%) as you do pulling 9 black marbles out one at a time, missing the white marble each time - which for previous ratios, was shown to be very difficult! - and the chance is still 10%.
I just can't wrap my head around this. I'm not stupid, I consider myself a learned man, but this just has me stumped.
Anyone out there want to put me out of my misery? :)
I've posted something similar to this before, and I never really got an answer that I could understand and made sense. It's something that's been bothering me for years, and it's so counter-intuitive that it drives me crazy. If someone with knowledge of statistics says "give up, just accept it, that's the way it is" - that's fine, I can do that. But if there's something deeper here, I'd love to hear about it.
Ok, so here we go.
This statistics problem takes the form of a game. You have a bag with ten marbles. The marbles can either be white or black, but there has to be ten total, and at least one white and at least one black. The way the game goes is this:
You put a combination of ten marbles into the bag, and draw one out. If it's white, the game is over. If it's black, you draw again, and keep drawing one at a time until you get a white one. That then ends the game.
For example, say there's 9 white marbles and 1 black marble. The chance of pulling a white marble on the first draw (ending the game outright) is 9/10, or 90%. The chance of pulling the black marble and then a white marble (to finish the game) is (1/10)*(9/9) = 10%.
Pretty simple stuff. Let's look at an example where there are 8 white marbles and 2 black marbles.
Chance of pulling white marble straight off: 8/10.
Chance of pulling one black, then a white: (2/10)*(8/9)=0.177
Change of pulling two blacks, then a white: (2/10)*(1/9)*(8/8)=0.022
This makes sense to me, logically. It's harder to draw two blacks then white, than it is to draw one black then white. Once you have one black marble out of the bag, you have to avoid all of those 8 white marbles floating around and find the one black marble. That's hard to do, avoiding all those white marbles.
I won't go through all the math because it's redundant, but this theme plays out as you go through 3 black / 7 white, 4 black / 6 white, 5 black / 5 white, 6 black / 4 white, 7 black / 3 white, and 8 black / 2 white. It becomes harder and harder to get all the black marbles out of the bag (avoiding the white ones, which would end the game) before finally hitting white. Hence the decreasing odds for emptying the bag of black marbles before hitting a white one.
This makes perfectly good sense to me, and it's backed up by the math if you'd care to draw it out, like I have above. It becomes *very* difficult to avoid hitting those white marbles until all the black ones are out.
However, something very strange happens when there are 9 black marbles and 1 white marble. Before we draw out the math, think about it: each time, we have to avoid hitting the white marble which becomes more and more difficult as the game continues. But let's look at the math:
Chance of hitting white straight away: 1/10 = 0.1.
Chance of hitting 1 black, then the white marble: (9/10)*(1/9)= 0.1.
Chance of hitting 2 blacks, then the white marble: (9/10)*(8/9)*(1/8)=0.1
......
......
Chance of hitting 5 blacks, then the white marble:
(9/10)*(8/9)*(7/8)*(6/7)*(5/6)*(1/5). As you can probably see, everything cancels, and we're left with odds of 1/10, or 0.1.
Taking it to the far extreme: chance of hitting all 9 blacks, missing the white marble *every single draw* - which in other starting marble ratios, was very difficult to do - before finally there's only one marble left in the bag, the white one, which is pulled out to end the game.
(9/10)*(8/9)*(7/8)*(6/7)*(5/6)*(4/5)*(3/4)*(2/3)*(1/2)*(1/1) = 1/10, or 0.1.
So you have an equal chance of pulling the one white marble out of ten in the bag straight off (10%) as you do pulling 9 black marbles out one at a time, missing the white marble each time - which for previous ratios, was shown to be very difficult! - and the chance is still 10%.
I just can't wrap my head around this. I'm not stupid, I consider myself a learned man, but this just has me stumped.
Anyone out there want to put me out of my misery? :)
via JREF Forum http://ift.tt/1lMszN5
Aucun commentaire:
Enregistrer un commentaire