p <- seq(0, 1, length.out = 1000)
plot(p, 0.5*dbeta(p, 31, 11) + 0.5*dbeta(p, 11, 31))
w <- c(0,0)
0.5*choose(4, 3)*beta(31+3, 11+1)/beta(31,11) -> w[1]
0.5*choose(4, 3)*beta(11+3, 31+1)/beta(11,31) -> w[2]
w
w <- w/sum(w)
w
plot(p, w[1]*dbeta(p, 31+3, 11+1) + w[2]*dbeta(p, 11+3, 31+1), type="l")