DataCamp | Probabilities | 1.3 | Addition Rule and Monty Hall
Jack LU
2/5/2021
Course: HarvardX PH125.3x | Data Science: Probability
Exercise 1. The Cavs and the Warriors
Two teams, say the Cavs and the Warriors, are playing a seven game championship series. The first to win four games wins the series. The teams are equally good, so they each have a 50-50 chance of winning each game.
If the Cavs lose the first game, what is the probability that they win the series?
# Assign a variable 'n' as the number of remaining games.
n <- 6
# Assign a variable `outcomes` as a vector of possible game outcomes, where 0 indicates a loss and 1 indicates a win for the Cavs.
outcomes <- c(0,1)
# Assign a variable `l` to a list of all possible outcomes in all remaining games. Use the `rep` function on `list(outcomes)` to create list of length `n`.
l <- rep(list(outcomes), n)
# Create a data frame named 'possibilities' that contains all combinations of possible outcomes for the remaining games.
possibilities <- expand.grid(l)
possibilities## Var1 Var2 Var3 Var4 Var5 Var6
## 1 0 0 0 0 0 0
## 2 1 0 0 0 0 0
## 3 0 1 0 0 0 0
## 4 1 1 0 0 0 0
## 5 0 0 1 0 0 0
## 6 1 0 1 0 0 0
## 7 0 1 1 0 0 0
## 8 1 1 1 0 0 0
## 9 0 0 0 1 0 0
## 10 1 0 0 1 0 0
## 11 0 1 0 1 0 0
## 12 1 1 0 1 0 0
## 13 0 0 1 1 0 0
## 14 1 0 1 1 0 0
## 15 0 1 1 1 0 0
## 16 1 1 1 1 0 0
## 17 0 0 0 0 1 0
## 18 1 0 0 0 1 0
## 19 0 1 0 0 1 0
## 20 1 1 0 0 1 0
## 21 0 0 1 0 1 0
## 22 1 0 1 0 1 0
## 23 0 1 1 0 1 0
## 24 1 1 1 0 1 0
## 25 0 0 0 1 1 0
## 26 1 0 0 1 1 0
## 27 0 1 0 1 1 0
## 28 1 1 0 1 1 0
## 29 0 0 1 1 1 0
## 30 1 0 1 1 1 0
## 31 0 1 1 1 1 0
## 32 1 1 1 1 1 0
## 33 0 0 0 0 0 1
## 34 1 0 0 0 0 1
## 35 0 1 0 0 0 1
## 36 1 1 0 0 0 1
## 37 0 0 1 0 0 1
## 38 1 0 1 0 0 1
## 39 0 1 1 0 0 1
## 40 1 1 1 0 0 1
## 41 0 0 0 1 0 1
## 42 1 0 0 1 0 1
## 43 0 1 0 1 0 1
## 44 1 1 0 1 0 1
## 45 0 0 1 1 0 1
## 46 1 0 1 1 0 1
## 47 0 1 1 1 0 1
## 48 1 1 1 1 0 1
## 49 0 0 0 0 1 1
## 50 1 0 0 0 1 1
## 51 0 1 0 0 1 1
## 52 1 1 0 0 1 1
## 53 0 0 1 0 1 1
## 54 1 0 1 0 1 1
## 55 0 1 1 0 1 1
## 56 1 1 1 0 1 1
## 57 0 0 0 1 1 1
## 58 1 0 0 1 1 1
## 59 0 1 0 1 1 1
## 60 1 1 0 1 1 1
## 61 0 0 1 1 1 1
## 62 1 0 1 1 1 1
## 63 0 1 1 1 1 1
## 64 1 1 1 1 1 1# Create a vector named 'results' that indicates whether each row in the data frame 'possibilities' contains enough wins for the Cavs to win the series.
results <- rowSums(possibilities)>=4
results## [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [13] FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE
## [25] FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE FALSE FALSE FALSE
## [37] FALSE FALSE FALSE TRUE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE
## [49] FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE
## [61] TRUE TRUE TRUE TRUE# Calculate the proportion of 'results' in which the Cavs win the series. Print the outcome to the console.
mean(results)## [1] 0.34375Exercise 2. The Cavs and the Warriors - Monte Carlo
Confirm the results of the previous question with a Monte Carlo simulation to estimate the probability of the Cavs winning the series after losing the first game.
Instructions
- Use the replicate function to replicate the sample code for B <- 10000 simulations.
- Use the sample function to simulate a series of 6 games with random, independent outcomes of either a loss for the Cavs (0) or a win for the Cavs (1) in that order. Use the default probabilities to sample.
- Use the sum function to determine whether a simulated series contained at least 4 wins for the Cavs.
- Use the mean function to find the proportion of simulations in which the Cavs win at least 4 of the remaining games. Print your answer to the console.
# The variable `B` specifies the number of times we want the simulation to run. Let's run the Monte Carlo simulation 10,000 times.
B <- 10000
# Use the `set.seed` function to make sure your answer matches the expected result after random sampling.
set.seed(1)
# Create an object called `results` that replicates for `B` iterations a simulated series and determines whether that series contains at least four wins for the Cavs.
results <- replicate(B, {
cavs_wins <- sample(c(0,1), 6, replace = TRUE)
sum(cavs_wins)>=4
})
# Calculate the frequency out of `B` iterations that the Cavs won at least four games in the remainder of the series. Print your answer to the console.
mean(results)## [1] 0.3371Exercise 3. A and B play a series - part 1
Two teams, A and B, are playing a seven series game series. Team A is better than team B and has a p> 0.5 chance of winning each game.
Instructions
- Use the function sapply to compute the probability, call it Pr of winning for
p <- seq(0.5, 0.95, 0.025). - Then plot the result
plot(p, Pr).
# Let's assign the variable 'p' as the vector of probabilities that team A will win.
p <- seq(0.5, 0.95, 0.025)
# Given a value 'p', the probability of winning the series for the underdog team B can be computed with the following function based on a Monte Carlo simulation:
prob_win <- function(p){
B <- 10000
result <- replicate(B, {
b_win <- sample(c(1,0), 7, replace = TRUE, prob = c(1-p, p))
sum(b_win)>=4
})
mean(result)
}
# Apply the 'prob_win' function across the vector of probabilities that team A will win to determine the probability that team B will win. Call this object 'Pr'.
Pr <- sapply(p, prob_win)
# Plot the probability 'p' on the x-axis and 'Pr' on the y-axis.
plot(p, Pr)
Exercise 4. A and B play a series - part 2
Repeat the previous exercise, but now keep the probability that team A wins fixed at p <- 0.75 and compute the probability for different series lengths. For example, wins in best of 1 game, 3 games, 5 games, and so on through a series that lasts 25 games.
Instructions - Use the seq function to generate a list of odd numbers ranging from 1 to 25. - Use the function sapply to compute the probability, call it Pr, of winning during series of different lengths. - Then plot the result plot(N, Pr).
# Given a value 'p', the probability of winning the series for the underdog team B can be computed with the following function based on a Monte Carlo simulation:
prob_win <- function(N, p=0.75){
B <- 10000
result <- replicate(B, {
b_win <- sample(c(1,0), N, replace = TRUE, prob = c(1-p, p))
sum(b_win)>=(N+1)/2
})
mean(result)
}
# Assign the variable 'N' as the vector of series lengths. Use only odd numbers ranging from 1 to 25 games.
N <- seq(1, 25, by=2)
# Apply the 'prob_win' function across the vector of series lengths to determine the probability that team B will win. Call this object `Pr`.
Pr <- sapply(N, prob_win)
# Plot the number of games in the series 'N' on the x-axis and 'Pr' on the y-axis.
plot(N, Pr)