There are two ways of choosing the coin. One is to pick a fair coin and the other is to pick the one with two heads.

Probability of selecting fair coin = 999/1000 = 0.999

Probability of selecting unfair coin = 1/1000 = 0.001

Selecting 10 heads in a row = Selecting fair coin * Getting 10 heads + Selecting an unfair coin

P (A) = 0.999 * (1/2)^5 = 0.999 * (1/1024) = 0.000976

P (B) = 0.001 * 1 = 0.001

P( A / A + B ) = 0.000976 / (0.000976 + 0.001) = 0.4939

P( B / A + B ) = 0.001 / 0.001976 = 0.5061

Probability of selecting another head = P(A/A+B) * 0.5 + P(B/A+B) * 1 = 0.4939 * 0.5 + 0.5061 = 0.7531