# Simulate one play of the Monty Hall game.  The player is assumed to
# choose a door at random to start.  If the argument of the function
# is TRUE, the player switches doors when offerred a choice; if FALSE,
# they don't switch.  The value returned is a list containing the
# doors chosen, opened, and with the car.

one.play <- function (switch)
{
    # Randomly choose which door has the car behind it,
    # and which door the player initially chooses.

    door.with.car <- sample (1:3, 1)
    door.chosen <- sample (1:3, 1)
 
    # Decide which door Monty Hall opens, choosing randomly from
    # among doors other than the one chosen and the one with the car.
    # This is done by repeatedly picking from among all three doors
    # until one that isn't excluded is chosen.

    repeat
    {   door.opened <- sample(1:3, 1)
        if (door.opened!=door.chosen && door.opened!=door.with.car)
        {   break
        }
    }

    # Have the player switch doors, if they're doing that.  They switch
    # to whichever door wasn't opened and isn't their current door.

    if (switch)
    {   for (i in 1:3)
        {   if (i!=door.chosen && i!=door.opened)
            {   door.chosen <- i
                break
            }
        }
    }

    # Return a list recording what happened.

    list (door.with.car=door.with.car, 
          door.chosen=door.chosen, 
          door.opened=door.opened)
}


# Simulate many plays of the Monty Hall game, recording how often the
# player wins the car.  The first argument is the number of plays to
# simulate.  The second argument says whether or not the player
# switches doors when given the chance.  The value returned is the
# fraction of times the player won.

many.plays <- function (n.plays, switch)
{
    # Simulate playing the game n.plays times, counting the number of wins.

    n.wins <- 0

    for (i in 1:n.plays)
    {   p <- one.play(switch)
        if (p$door.chosen == p$door.with.car)
        {   n.wins <- n.wins + 1
        }
    }

    # Return the fraction of plays when the player wins the car.

    n.wins / n.plays
}