Sampling distribution of floods. See https://tonyladson.wordpress.com/2020/06/02/sampling-distribution-of-the-1-flood/

Flood_sampling_distribution.R

library(tidyverse)

#pbinom(q, size, prob, lower.tail = TRUE, log.p = FALSE)

# Probability  of 1, 1% flood in 100 years

dbinom(1, size = 100, prob = 0.01, log = FALSE)
choose(100,1) *0.01^1*0.99^99


# Probability of 5 or more

1-pbinom(4, size = 100, 0.01)



# Create data frame for plotting

sampling_dist <- expand_grid(num_events = 0:10, years = 100, event_prob = 0.01)

sampling_dist <- sampling_dist %>% 
  rowwise() %>% 
  mutate(prob_num = dbinom(x = num_events, size = years, prob = event_prob))
  


p <- sampling_dist %>% 
  ggplot(aes(x = num_events, y = prob_num)) +
  geom_col(fill = 'blue', colour = 'black') +
  scale_x_continuous(name = 'Number of events', breaks = 0:10) +
  scale_y_continuous(name = 'Probability of the number of events', limits = c(0, 0.4)) +
  theme_gray(base_size = 7)


ggsave(file.path('../figures','Sampling_dist_1pc_100y.png'), p, width = 4, height = 3)


# Expected number of floods
# Sum x * Pr(x)

sampling_dist %>% 
  mutate(mean_component = num_events * prob_num)  %>% 
  ungroup() %>% 
  dplyr::summarize(sum(mean_component), n = n())

# or
100*0.01
  

##################

# AEP = 2%

sampling_dist <- expand_grid(num_events = 0:10, years = 100, event_prob = 0.02)

sampling_dist <- sampling_dist %>% 
  rowwise() %>% 
  mutate(prob_num = dbinom(x = num_events, size = years, prob = event_prob))



sampling_dist %>% 
  ggplot(aes(x = num_events, y = prob_num)) +
  geom_col(fill = 'blue', colour = 'black') +
  scale_x_continuous(name = 'Number of events', breaks = 0:10) +
  scale_y_continuous(name = 'Probability of the number of events', limits = c(0, 0.4))




p <- sampling_dist %>% 
  ggplot(aes(x = num_events, y = prob_num)) +
  geom_col(fill = 'blue', colour = 'black') +
  scale_x_continuous(name = 'Number of events', breaks = 0:10) +
  scale_y_continuous(name = 'Probability of the number of events', limits = c(0, 0.4)) +
  theme_gray(base_size = 7)

ggsave(file.path('../figures','Sampling_dist_2pc_100y.png'), p, width = 4, height = 3)





# -------------------------------------------------------------------------

#Change in Risk bewteen a 1% and 2% flood


# Probability of 1 or more floods
# Comparing 2% AEP to 1% AEP

1-pbinom(0, size = 100, prob = 0.02, log = FALSE) #0.867
1-pbinom(0, size = 100, prob = 0.01, log = FALSE) #0.633


(1-pbinom(0, size = 100, prob = 0.02, log = FALSE))/(1-pbinom(0, size = 100, prob = 0.01, log = FALSE))



# Probability of 5 or more floods
# Comparing 2% AEP to 1% AEP

(1-pbinom(4, size = 100, prob = 0.02, log = FALSE))

(1-pbinom(4, size = 100, prob = 0.02, log = FALSE))/(1-pbinom(4, size = 100, prob = 0.01, log = FALSE))

#14.8

# Compare 1% with 0.05%

1-pbinom(0, size = 100, prob = 0.005, log = FALSE) #0.39
1-pbinom(0, size = 100, prob = 0.01, log = FALSE) #0.633

(1-pbinom(4, size = 100, prob = 0.005, log = FALSE))

(1-pbinom(4, size = 100, prob = 0.005, log = FALSE))/(1-pbinom(4, size = 100, prob = 0.01, log = FALSE))

1/((1-pbinom(4, size = 100, prob = 0.005, log = FALSE))/(1-pbinom(4, size = 100, prob = 0.01, log = FALSE)))