discrete distributions

library(tidyverse)

single_trial_prob <- .5

num_trials <- 2

0:num_trials ->
  possible_outcomes

dbinom(x = possible_outcomes, 
       size = 2,
       prob = single_trial_prob) ->
  probs

tibble(possible_outcomes, probs) %>% 
  ggplot() + 
  aes(x = possible_outcomes) +
  scale_x_continuous(breaks = possible_outcomes) +
  aes(y = probs) + 
  geom_point() + 
  aes(xend = possible_outcomes,
      yend = 0) + 
  geom_segment(lty = "dotted") + 
  labs(title = "Binomial Distribution of for ten events where probability of success for each event is .85  (success/failure probability is independent)" %>% 
         str_wrap()) + 
  geom_text(aes(label = probs %>% round(4)),
            nudge_y = .01) + 
  annotate(geom = "text", 
           x = 1.5,
           y = .2,
           label = "nCk(pi^k)*(1-pi)^(n-k)",
           parse = TRUE,
           size = 5)

Board Problems for Lesson 27: Binomial, Geometric, and Normal Distributions

1. Michael Jordan, Hall of Fame basketball player, made about 85% of his free throws during his prime years with the Chicago Bulls. During a free throw contest, he will attempt 10 shots. Assume all of his free throws are independent of one another.

1. What is the probability that Jordan makes exactly 7 of his attempts?
2. What is the probability that Jordan makes more than 7 free throws?
3. What is the probability that Jordan makes at least 7 of his attempts?

library(tidyverse)

single_trial_prob <- .85

num_trials <- 10

0:num_trials ->
  possible_outcomes

dbinom(x = possible_outcomes, 
       size = 10,
       prob = single_trial_prob) ->
  probs

tibble(possible_outcomes, probs) %>% 
  ggplot() + 
  aes(x = possible_outcomes) +
  scale_x_continuous(breaks = possible_outcomes) +
  aes(y = probs) + 
  geom_point() + 
  aes(xend = possible_outcomes,
      yend = 0) + 
  geom_segment(lty = "dotted") + 
  labs(title = "Binomial Distribution of for ten events where probability of success for each event is .85  (success/failure probability is independent)" %>% 
         str_wrap()) + 
  geom_text(aes(label = probs %>% round(4)),
            nudge_y = .01) + 
  annotate(geom = "text", 
           x = 1.5,
           y = .2,
           label = "nCk(pi^k)*(1-pi)^(n-k)",
           parse = TRUE,
           size = 5)

# what is the probability of at least eight successes in ten trials
# when each trial is chance event with probability of .8
8:10 %>% 
  dbinom(x = ., 
         size = 10,
         prob = .8) %>% 
  sum()

## [1] 0.6777995

# or probabilities associated with single outcome
dbinom(x = 8, size = 10, .8) + 
  dbinom(9, 10, .8) +
  dbinom(10, 10, .8)

## [1] 0.6777995

# what is the probability of exactly 5 successes in 10 trials
# when each trial is chance event with probability .5 success
dbinom(5, 10, .5)

## [1] 0.2460938

---

\[\binom{n}k(\pi^k)(1-\pi)^{n-k}\]

–

Where n choose k (first part)

\[\frac{n!}{(n-k)!k!} \]

---

choose(5,2) *
  .341^2 *
  (1 - .341)^3

## [1] 0.332786

dbinom(2, 5, .341)

## [1] 0.332786

2. Historically, a Soldier hits a target at 50 meters 80% of the time. Assume shots are independent of one another.

1. What is the probability that the Soldier misses on the first two shots?
2. What is the probability that the Soldier misses a few in a row, and then hits the target on the 4th shot?
3. What is the probability that the Soldier requires at least 4 shots before hitting the target?

library(tidyverse)

single_trial_prob <- .8

num_trials <- 10

0:num_trials ->
  possible_outcomes

dgeom(x = possible_outcomes, 
       prob = single_trial_prob) ->
  probs

tibble(possible_outcomes, probs) %>% 
  ggplot() + 
  aes(x = possible_outcomes) +
  scale_x_continuous(breaks = possible_outcomes) +
  aes(y = probs) + 
  geom_point() + 
  aes(xend = possible_outcomes,
      yend = 0) + 
  geom_segment(lty = "dotted") + 
  labs(title = "Geometric Distribution success by events where probability of success for each event is .8  (success/failure probability is independent)" %>% 
         str_wrap()) + 
  geom_text(aes(label = probs %>% round(4)),
            nudge_y = .01) + 
  annotate(geom = "text", 
           x = 1.5,
           y = 1,
           label = "(pi^k)*(1-pi)^(n-k)",
           parse = TRUE,
           size = 5)

---

3. The weekly amount of diesel fuel your unit uses was observed, over a long period of time, to be approximately normally distributed with a mean of 400 gallons and a variance of 400 gallons^2.

1. If 420 gallons are budgeted for next week, what is the probability that the actual amount consumed will be within the budget?
2. If 450 gallons are budgeted for next week, what is the probability that the actual amount consumed will exceed the budgeted amount?
3. What is the probability your unit uses between 390 and 425 gallons of fuel

library(ggxmean)
ggxmean:::stamp_space() +
  stamp_normal_dist(sd = 400^.5, mean = 400) + 
  stamp_normal_dist_zlines(sd = 400^.5, # sd = 20
                           mean = 400,
                           height = 20) ->
basenormal

## Warning: Ignoring unknown parameters: outline.type

basenormal

basenormal +
  geom_vline(xintercept = 420, color = "green")

basenormal + 
  geom_vline(xintercept = 450, color = "blue")

basenormal + 
  geom_vline(xintercept = c(390, 425), 
             linetype = "dotted",
             color = "red")