n <- 12*10^6
p <- 1/n
1 - ((1-p)^(n-1))/(1 - (1-p)^n)

## [1] 0.4180233

ps <- 1 - 0.9*0.95*0.85

ps1 <- 0.1
ps2 <- 0.9*0.05
ps3 <- 0.9*0.95*0.15

p <- c(ps1, ps2, ps3)/ps
p

## [1] 0.3659652 0.1646844 0.4693504

sum(p)

## [1] 1