Consider a box containing 10 balls: 4 blue and 6 red, as displayed in the following figure. Assume you pick a 1st ball, then a 2nd ball (no replacement).
Find the probability that:
both balls are blue.
both balls are red.
1st is blue, 2nd is red.
1st is red, 2nd is blue.
2nd is blue, given 1st is red.
2nd is blue, given 1st is blue.
1st is blue, given 2nd is red.
at least one of them is blue.
both are of the same color.
library(tidyverse)Refer to the box of the preceding problem. The following code allows you to simulate drawing 2 balls from the box 100 times—via replicate(). In order to use tidyverse functions, we need to reshape the simulations output into a data.frame.
set.seed(2024)
box = c(rep("red", 6), rep("blue", 4))
# simulate drawing 2 balls, 100 times
simulations = replicate(
n = 100,
expr = sample(box, size = 2))
# table output (as a data frame)
rownames(simulations) = c("first", "second")
balls_df = data.frame(t(simulations))
head(balls_df) first second
1 red red
2 red blue
3 blue red
4 red blue
5 blue red
6 red redSay we want to approximate probability of getting a first blue ball.
# proportion: 1st ball is blue
balls_df |>
summarize(prop_1blue = mean(first == "blue")) prop_1blue
1 0.44What if we we want to approximate probability of getting the first ball blue, and the second ball red?
# proportion: 1st is blue, 2nd is red
balls_df |>
summarize(prop_1blue_2red = mean(first == "blue" & second == "red")) prop_1blue_2red
1 0.3Change the value of set.seed() and increase the number of simulations to 1000. Write R pipelines to approximate the following probabilities. How do these approximations compare to the probabilities you calculated?
balls_df |>
summarize(mean(first == "blue" & second == "blue"))# b) 1st is red, 2nd is blue
balls_df |>
summarize(mean(first == "red" & second == "blue"))# c) both are red
balls_df |>
summarize(mean(first == "red" & second == "red"))# d) 1st is blue OR 2nd is blue
balls_df |>
summarize(mean(first == "blue" | second == "blue"))# e) 2nd is blue, given that 1st is red
balls_df |>
filter(first == "red") |>
summarize(mean(second == "blue"))# f) 2nd is blue, given that 1st is blue
balls_df |>
filter(first == "blue") |>
summarize(mean(second == "blue"))# g) 1st is blue, given that 2nd is red
balls_df |>
filter(second == "red") |>
summarize(mean(first == "blue"))
Suppose that 35% of students in STAT 101 are from Southern California, 40% are from Northern California, and 25% are from outside California. What is the probability that a student selected at random:
Will be from Southern CA?
Will not be from Northern CA?
Will be from Southern or Northern CA?
Will be from Northern California, given that they are not from Southern CA?
Will be from Northern California, given that they are not from outside CA?
Suppose we draw 2 tickets at random without replacement from a box with tickets marked {1, 2, 3, . . . , 9}. Find the probability that:
The first ticket drawn is labeled with an even number.
The second ticket drawn is labeled with an even number.
Both tickets drawn are labeled with an even number.
At least one of the tickets drawn is labeled with an even number.
The first ticket drawn is labeled with a prime number (recall that 1 is not a prime).
The second ticket drawn is labeled with a prime number.
Both tickets drawn are labeled with a prime number.
At least one of the tickets drawn is labeled with a prime number.
Match the terms (a-f) with their definitions (i-vi)
Terms
Definitions
You roll a red die and a blue die, both fair and independent. Find the probability that:
The red die and the blue die both roll 1’s.
The red die and the blue die both roll the same number.
The number on the red die is bigger than the one on the blue die.
The red die rolls a 3.
At least one die rolls a 3.
Exactly one die rolls a 3.
The smallest number on either die is 4.
The largest number on either die is 4.
At least one of the dice rolls a number greater than 4.
Exactly one of the dice rolls a number greater than 4.
The numbers on the dice sum to 7.
The numbers on the dice sum to 4.
A students’ club has 90 members. Fifty are Statistics majors and fifty are Applied Math (A.M.) majors. There are some that are double majoring in Stats and A.M. What is the probability that a member selected at random:
Is a double major?
Is a Stats major but not an A.M. major?
Is an A.M. major but not a Stats major?
Suppose that A and B are independent events with \(P(A)\) = 0.7, and \(P(B^c)\) = 0.4. Find the following probabilities:
\(P(A^c)\)
\(P(B)\)
\(P(B \text{ and } A)\)
\(P(A \text{ or } B)\)
\(P(A^c \text{ and } B)\)
\(P(B | A)\)