Write a program to toss a coin 10,000 times. Let Sn be the number of heads in the first n tosses. Have your program print out, after every 1000 tosses, Sn − n/2. On the basis of this simulation, is it correct to say that you can expect heads about half of the time when you toss a coin a large number of times?
Solution:
n <- 1000 # Number of tosses equal to 1000
Sn <- c() # Stores the number of heads
sum_heads <- 0 # Stores the sum of the number of heads
n <- c() # Stores the number of tosses
for(i in 1:10){
simulation <- sample(0:1, 1000, rep = T) # Tosses coin 1000 times
sum_heads <- sum_heads + sum(simulation == 1) # Gets the sum of heads
Sn <- c(Sn, sum_heads / (1000 * i)) # Gets the number of heads
n <- c(n, 1000 * i) # Gets the number of tosses
}
data_frame <- data.frame(n, Sn) # Dataframe to show the number of tosses and number of heads
data_frame## n Sn
## 1 1000 0.4990000
## 2 2000 0.5035000
## 3 3000 0.5103333
## 4 4000 0.4955000
## 5 5000 0.4964000
## 6 6000 0.4966667
## 7 7000 0.4974286
## 8 8000 0.4983750
## 9 9000 0.4962222
## 10 10000 0.4995000For my program, it runs a simulation of 1000 coin tosses, where heads is a 1 and tails is a 0, and loops through 10 times to show 10000 tosses. Each time it loops, the number of heads, the sum of heads, and the number of tosses are being stored. I created a dataframe to display every 1000 tosses and the number of heads for every 1000 tosses. Based on the simulation and the dataframe I displayed, it is correct to say that you can expect heads about half of the time when you toss a coin a large number of times. This is because you can see as the tosses go from 1000 to 10000, the number of heads are within 1/2 of the tosses.