Imagine you draw two balls from a box containing colored balls. You either replace the first ball before you draw the second or you leave the first ball out of the box when you draw the second ball.
Under which situation are the two draws independent of one another? Remember that two events A and B are independent if Pr(A and B) = Pr(A) * P(B).
ANSWER: You do replace the first ball before drawing the next
Say you’ve drawn 5 balls from the a box that has 3 cyan balls, 5 magenta balls, and 7 yellow balls, with replacement, and all have been yellow.
What is the probability that the next one is yellow?
p_yellow as the probability of choosing a yellow ball on the first draw.p_yellow, calculate the probability of choosing a yellow ball on the sixth draw.cyan <- 3
magenta <- 5
yellow <- 7
# Assign the variable 'p_yellow' as the probability that a yellow ball is drawn from the box.
p_yellow <- yellow/(cyan+magenta+yellow)
# Using the variable 'p_yellow', calculate the probability of drawing a yellow ball on the sixth draw. Print this value to the console.
p_yellow## [1] 0.4666667If you roll a 6-sided die once, what is the probability of not seeing a 6? If you roll a 6-sided die six times, what is the probability of not seeing a 6 on any of those rolls?
# Assign the variable 'p_no6' as the probability of not seeing a 6 on a single roll.
p_no6 <- 1 - (1/6)
# Calculate the probability of not seeing a 6 on six rolls using `p_no6`. Print your result to the console: do not assign it to a variable.
p_no6^6## [1] 0.334898Two teams, say the Celtics and the Cavs, are playing a seven game series. The Cavs are a better team and have a 60% chance of winning each game.
What is the probability that the Celtics win at least one game? Remember that the Celtics must win one of the first four games, or the series will be over!
# Assign the variable `p_cavs_win4` as the probability that the Cavs will win the first four games of the series.
p_cavs_win4 <- .6^4
# Using the variable `p_cavs_win4`, calculate the probability that the Celtics win at least one game in the first four games of the series.
1 - p_cavs_win4## [1] 0.8704Create a Monte Carlo simulation to confirm your answer to the previous problem by estimating how frequently the Celtics win at least 1 of 4 games. Use B <- 10000 simulations.
The provided sample code simulates a single series of four random games, simulated_games.
Instructions
replicate function for B <- 10000 simulations of a four game series. The results of replicate should be stored to a variable named celtic_wins.mean function on celtic_wins to find the proportion of simulations that contain at least one win for the Celtics out of four games.# This line of example code simulates four independent random games where the Celtics either lose or win. Copy this example code to use within the `replicate` function.
simulated_games <- sample(c("lose","win"), 4, replace = TRUE, prob = c(0.6, 0.4))
# 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 `celtic_wins` that replicates two steps for B iterations: (1) generating a random four-game series `simulated_games` using the example code, then (2) determining whether the simulated series contains at least one win for the Celtics. Put these steps on separate lines.
celtic_wins <- replicate(B, {
simulated_games <- sample(c("lose","win"), 4, replace = TRUE, prob = c(0.6, 0.4))
any(simulated_games=="win")
})
# Calculate the frequency out of B iterations that the Celtics won at least one game. Print your answer to the console.
mean(celtic_wins)## [1] 0.8757