We’ll solve the below problems using simulation. We’ll set up a uniform distirbution and simulate 100 tosses of a fair coin. We’ll do this 10K times and then count up how many times we met our criteria out of 10K to get our probability.
Let \({ S }_{ 100 }\) be the number of heads that turn up in 100 tosses of a fair coin. Use the Central Limit Theorem to estimate
(a)\({ P(S }_{ 100 }\le 45)\):
count <- c()
for (i in seq(1,10000)) {
z <- runif(100, min = 0, max = 1)
count <- c(count, length(z[z>=.5]))
}
final_count <- length(count[count<=45])
final_count/10000## [1] 0.1832(b)\({ P(45<S }_{ 100 }<55)\):
count <- c()
for (i in seq(1,10000)) {
z <- runif(100, min = 0, max = 1)
count <- c(count, length(z[z>=.5]))
}
final_count <- length(count[count>45 & count <55])
final_count/10000## [1] 0.6376(c)\({ P(S }_{ 100 }>63)\):
count <- c()
for (i in seq(1,10000)) {
z <- runif(100, min = 0, max = 1)
count <- c(count, length(z[z>=.5]))
}
final_count <- length(count[count>63])
final_count/10000## [1] 0.0024(d)\({ P(S }_{ 100 }<57)\):
count <- c()
for (i in seq(1,10000)) {
z <- runif(100, min = 0, max = 1)
count <- c(count, length(z[z>=.5]))
}
final_count <- length(count[count<57])
final_count/10000## [1] 0.9035