ages = seq(90, 101)
probs = c( 0.07, 0.08, 0.09, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.07, 0.05, 0.04 )

Abar = sum(ages * probs)

P_of_n = function(n){
  P = (A/r)*(1 - 1/(1+r)^n )
}

# Part (b):
n = Abar - 90.0 # how many payments will be made on average
A = 10000
r = 0.08

P_part_b = P_of_n(n)
print(P_part_b)

# Part (c):
P_of_n_values = sapply(0:11, P_of_n)
sum( P_of_n_values * probs )