Since your very first selection has 50/50 probability of red vs. black, that means the number of red and black balls are equal. r = b

Once you’ve picked out two balls of the same color (assume they are black), then the probability of picking consecutive red balls in your next two draws = (r/(r+b-2))*(r-1)/(r+b-3) = 0.5

Substitute b in for r and do the multiplication: (r^2 – r)/(4r^2 – 10r +6) = 0.5

Do some algebra to isolate everything on one side: 0 = r^2 -4r + 3

0 = (r-3)(r-1) –> r=3 or r=1. But r cannot = 1 because you must be able to pick out 2 black balls at the onset. Therefore r=b=3, so the urn has 6 balls.

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