Question

Simulate with Rstudio and also make sure to compare by solving by hand(exact calculation) Show all work.

What is the expected value of a binomial distribution where 25 coins are flipped, each having a 30% chance of heads? • Compare your simulation with the exact calculation.

What is the variance of a binomial distribution where 25 coins are flipped, each having a 30% chance of heads? • Compare your simulation with the exact calculation.

Example:If ? ~???????? 10, . 5 , what is the E[?]? using calculation E ? = 5.

• Simulation: run the experiment 100,000 times.

• flips <- rbinom (100000, 10, .5 )

• mean (flips): the average number of heads

Answer 1

# Generate 100 occurrences of flipping 10 coins, each with 30% probability
rbinom(100,10,.3)

FALSE   [1] 2 2 0 4 4 5 3 2 3 2 2 3 4 4 3 4 6 1 2 1 1 3 3 2 4 3 2 3 3 2 3 4 2 2 4
FALSE  [36] 0 2 5 2 2 6 3 1 3 1 2 1 2 3 2 3 2 1 2 5 3 4 3 2 3 3 6 3 2 1 3 1 5 2 2
FALSE  [71] 4 5 0 3 1 3 3 4 0 3 3 4 3 4 5 5 4 1 1 4 1 4 2 4 3 4 3 2 1 2

# Calculate the probability that 2 are heads using dbinom
dbinom(2,10,.3)

FALSE [1] 0.2334744

# Confirm your answer with a simulation using rbinom
mean(rbinom(10000,10,.3)==2)

FALSE [1] 0.2391

# Calculate the probability that at least five coins are heads
1-pbinom(4,10,.3)

FALSE [1] 0.1502683

# Confirm your answer with a simulation of 10,000 trials
mean(rbinom(10000,10,.3)>=5)

FALSE [1] 0.1497

# Here is how you computed the answer in the last problem
mean(rbinom(10000, 10, .3) >= 5)

FALSE [1] 0.1437

# Try now with 100, 1000, 10,000, and 100,000 trials
mean(rbinom(100, 10, .3) >= 5)

FALSE [1] 0.15

mean(rbinom(1000, 10, .3) >= 5)

FALSE [1] 0.156

mean(rbinom(10000, 10, .3) >= 5)

FALSE [1] 0.1501

mean(rbinom(100000, 10, .3) >= 5)

FALSE [1] 0.14969

# Calculate the expected value using the exact formula
25*.3

FALSE [1] 7.5

# Confirm with a simulation using rbinom
mean(rbinom(10000,25,.3))

FALSE [1] 7.5268

# Calculate the variance using the exact formula
25*.3*(1-.3)

FALSE [1] 5.25

# Confirm with a simulation using rbinom
var(rbinom(10000,25,.3))

FALSE [1] 5.278581

# Simulate 100,000 flips of a coin with a 40% chance of heads
A <- rbinom(100000, 1, .4)

# Simulate 100,000 flips of a coin with a 20% chance of heads
B <- rbinom(100000, 1, .2)

# Estimate the probability both A and B are heads
mean(A & B)

FALSE [1] 0.07967

# You've already simulated 100,000 flips of coins A and B
A <- rbinom(100000, 1, .4)
B <- rbinom(100000, 1, .2)

# Simulate 100,000 flips of coin C (70% chance of heads)
C <- rbinom(100000, 1, .7)

# Estimate the probability A, B, and C are all heads
mean(A&B&C==1)

FALSE [1] 0.05593