A bag contains an unknown number of red balls and yellow balls. When N balls are drawn at random (without replacement) the probability that they are all yellow is 1/2. The number of balls in the bag is the minimum for this to happen.
If the first N balls were all yellow, what is the probability that the next ball drawn is red?
Express the probability as a function of N.
(In reply to
re(2): some exceptions by Brian Smith)
I decided to explore the other outlier (N,R,Y) = (6,2,19). This makes for an R=2 family.
Then the equation implied by the problem becomes 2*y*(y-1)*...*(y-n+3)*(y-n+2)*(y-n+1) = (y+2)*(y+1)*y*...*(y-n+3).
This simplifies to 2*(y-n+2)*(y-n+1) = (y+2)*(y+1).
Let y-n+2=Z and y+2=W, then this reduces further to (2W-1)^2 - 2*(2Z-1)^2 = -1.
This is the same Pell-type equation from my previous post. Then we get values for 2Z-1 from A001653. Also, I omitted this last time but the sequence of values for 2W-1 are A002315.
Then from (2W-1,2Z-1) = (5,7), (29,41), (169,239), (985, 1393), etc we can get (N,Y) = (1,2), (6,19), (35,118), (204,695), etc
This is another family of (non-minimum) solutions. I guess there is one more open problem: Are there any solutions with all three of R,N,Y greater than 2?
re(3): some exceptions
