Simulation and Exact Probability

A fair six-sided die is rolled six times. If the face numbered "j" is the outcome on the j-th roll we say "a match has occurred." The experiment is declared "success" if at least one match occurs during the six trials.

1. Simulate this experiment 500 times, and evaluate the proportion of "success" event.
2. Compute the exact probability of "success,"

   exact.prob = 1 - (5/6)^6
   

   and compare it with the evaluation via simulation.

Here we have to simulate a slightly complicated experiment: Generate 6 trials of rolling a fair six-sided die, and find out whether the j-th attempt results in the exact same number j. We can write all the tasks of experiment by defining a function

is.matched()

The function is.match(outcome) returns the value "1" if at least one match is found in "outcome" of the six trials. Otherwise, it returns "0."

is.matched = function(outcome){
  match.status = 0;
  for(j in 1:6){
    if(outcome[j] == j)
      match.status = 1;
  }
  match.status
}

The function sample(6,size=6,replace=T) randomly generates numbers drawn from 1 to 6 for 6 times (size=6). Then use it to simulate the experiment repeatedly for n = 500 times. It evaluates the proportion of "success" event, and creates "success.rate"

n = 500;
success = NULL;
success.rate = NULL;
prob = rep(1/6,6);
for(i in 1:n){
  outcome = sample(6,size=6,replace=T);
  success[i] = is.matched(outcome);
  success.rate[i] = mean(success);
}

Here we use plot() and lines() to visualize the success rate in series. Compute the exact probability for "winning the game," and compare it with the evaluation via simulation.

exact.prob = 1 - (5/6)^6;
plot(c(1,n), rep(exact.prob,2), ylab='Fraction of successes', type='l', lty=2);
lines(1:n, success.rate, col='blue');
print(success.rate[n])

Sample R code. You can download diegame.R, and run it by

source(file.choose())

where file.choose() lets you choose the R code.

Programming Note. Here the results of repeated experiment are stored in the data variable

success

where the event of successful match is indicated by 1; otherwise it is 0.

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