Originally posted by wolfgang59
Could you elaborate?
I didnt have an answer to this problem; 20 mins seems a sensible guess but when you consider the quickest time they could all fall is 10 minutes (lets say that a bottle falls or not after each 1 min period) and the longest is infinite does the average come out at 20 minutes?
In 1 minute, there is a 1/2 chance of 10 bottles left, and a 1/2 chance of 9 bottles left. (a+b)^1 = 1
*a + 1
in two minutes, there is a 1/4 chance of 10 bottles left, a 2/4 chance of 9 bottles left, and a 1/4 chance of 8 bottles left. (a+b)^2 = 1
*a^2 + 2
*ab + 1
In three minutes: (a+b)^3 = 1
*a^3 + 3
*a^2b + 3
*ab^2 + 1
So there is a 1/8 chance of 10 bottles
3/8 chance of 9 bottles
3/8 chance of 8 bottles
1/8 chance of 7 bottles
As you can see, there will always be an equal sum of coefficients above and below the median. Also, as you'll notice, the exponent of "b" can represent the number of bottles that has fallen. so when the coeffient of b^10 is the maximum coefficient, there there is a >=50% chance that all bottles has fallen
*edit* you can use this link to help you expand the polynomials.
You will notice that since this is a binary expansion, the sum of all of the coefficients will always be 2^n