Employ R to use random number generation and simulations to verify theoretical probabilities.

Sure, here is a simple example of how you can use R to generate random numbers and simulate to verify theoretical probabilities.

Let's say we want to verify the theoretical probability of getting a head when flipping a fair coin (which is 0.5). We can simulate this in R as follows:

1. First, we need to set up the simulation. We can use the set.seed() function to make our results reproducible:

set.seed(123)

2. Next, we generate a large number of random numbers that are either 0 (representing tails) or 1 (representing heads). We can use the sample() function to do this:

flips <- sample(c(0, 1), size = 10000, replace = TRUE)

In this code, c(0, 1) creates a vector of the two possible outcomes, size = 10000 means we want to generate 10000 random numbers, and replace = TRUE means we're sampling with replacement, which is appropriate for independent events like coin flips.

3. Now, we can calculate the proportion of heads in our simulated flips, which is an estimate of the probability of getting a head:

prob_head <- sum(flips) / length(flips)
print(prob_head)

This code sums up the flips vector (which adds 1 for each head and 0 for each tail), and then divides by the total number of flips to get the proportion of heads.

4. Finally, we can compare our simulated probability with the theoretical probability:

theoretical_prob <- 0.5
print(abs(theoretical_prob - prob_head))

This code calculates the absolute difference between the theoretical probability and our simulated probability. If our simulation is accurate, this difference should be close to 0.

Remember, the accuracy of the simulation increases with the number of trials (in this case, the number of coin flips). So, if you want a more accurate estimate, you can increase the size parameter in the sample() function.