sim3.6

p_defective <- 3 / 75

probA <- dgeom(5, prob = p_defective)
probA

[1] 0.03261491

probB <- 1 - pgeom(7, prob = p_defective)
probB

[1] 0.7213896

expC <- (1 - p_defective) / p_defective
expC

[1] 24

expY <- 1 / p_defective
expD_cents <- 45 * expY
expD_dollars <- expD_cents / 100
expY

[1] 25

expD_cents

[1] 1125

expD_dollars

[1] 11.25

set.seed(123)
num_sim <- 10000

sim_X <- rgeom(num_sim, prob = p_defective)

#sim_X

sim_Y <- sim_X + 1

#sim_Y

sim_probA <- mean(sim_Y == 6)
sim_probB <- mean(sim_X >= 8)
sim_expC  <- mean(sim_X)
sim_expY  <- mean(sim_Y)
sim_expD_cents <- 45 * sim_expY

data.frame(
  Quantity = c("A: P(Y=6)", "B: P(X>=8)", "C: E(X) good bulbs",
               "D: E(Y) bulbs tested", "D: Expected cost (cents)"),
  Theoretical = c(probA, probB, expC, expY, expD_cents),
  Simulated   = c(sim_probA, sim_probB, sim_expC, sim_expY, sim_expD_cents)
)

                  Quantity  Theoretical Simulated
1                A: P(Y=6) 3.261491e-02    0.0276
2               B: P(X>=8) 7.213896e-01    0.7276
3       C: E(X) good bulbs 2.400000e+01   24.3324
4     D: E(Y) bulbs tested 2.500000e+01   25.3324
5 D: Expected cost (cents) 1.125000e+03 1139.9580

p_strike <- 0.20
r_strikes <- 3

probA2 <- dnbinom(4, size = r_strikes, prob = p_strike)
probA2

[1] 0.049152

probB2 <- 1 - pnbinom(6, size = r_strikes, prob = p_strike)
probB2

[1] 0.7381975

X_40 <- qnbinom(0.40, size = r_strikes, prob = p_strike)
Y_40 <- X_40 + r_strikes
Y_40

[1] 12

expY2 <- r_strikes / p_strike
expY2

[1] 15

set.seed(123)
num_sim <- 10000

sim_X2 <- rnbinom(num_sim, size = r_strikes, prob = p_strike)

#sim_X2

sim_Y2 <- sim_X2 + r_strikes

#sim_Y2

sim_probA2 <- mean(sim_Y2 == 7)
sim_probB2 <- mean(sim_Y2 >= 10)
sim_Y_40   <- quantile(sim_Y2, 0.40, type = 1)
sim_expY2  <- mean(sim_Y2)

data.frame(
  Quantity = c("A: P(Y=7)", "B: P(Y>=10)", "C: ~40th pct wells",
               "D: E(Y) wells drilled"),
  Theoretical = c(probA2, probB2, Y_40, expY2),
  Simulated   = c(sim_probA2, sim_probB2, sim_Y_40, sim_expY2)
)

               Quantity Theoretical Simulated
1             A: P(Y=7)   0.0491520    0.0498
2           B: P(Y>=10)   0.7381975    0.7291
3    C: ~40th pct wells  12.0000000   12.0000
4 D: E(Y) wells drilled  15.0000000   14.8633