Consider the disaster risk decision tree model.
(a) Derive a formula to represent the amount that the probability of all suppliers being disrupted simultaneously, P(n), will increase if the super-event probability S is doubled.
(b) Test your formula by computing the amount of increase if the original S equals 1% and there are two suppliers, each with U = 4%.

(a) Formula (S11 - 1) can be rewritten as: P(n) = S + Uⁿ - SUⁿ
If S doubles: P(n) = 2S + Uⁿ - 2SUⁿ
Subtracting the first equation from the second yields an increase of:
S - SU^n, or S(1 - Uⁿ)
(b) The original P(2) = .01 + (1 - .01)(.04²) = .01 + .001584 = .011584
If S doubles to .02, P(2) = .02 + (1 - .02)(.04²) = .02 + .001584 = .021568
The increase = .021568 - .011584 = .009984 = .01(1 - .04²)