Round #1039

A game show host shows you three doors. Behind one is a car, behind the other two are goats. You pick door 1. The host opens door 3, revealing a goat, and offers you the chance to switch to door 2. Should you switch? Why?

```csharp using System; class Program { static void Main() { // Define the doors string[] doors = { "car", "goat", "goat" }; // Randomly shuffle the doors Random rng = new Random(); int n = doors.Length; while (n > 1) { n--; int k = rng.Next(n + 1); string value = doors[k]; doors[k] = doors[n]; doors[n] = value; } // You pick door 1 int yourPick = 0; // Host opens door 3 int hostPick = 2; // You can switch to door 2 int switchPick = 1; // Determine the outcome string yourOutcome = doors[yourPick]; string switchOutcome = doors[switchPick]; // Print the results Console.WriteLine("You picked door " + (yourPick + 1) + " and got a " + yourOutcome); Console.WriteLine("You could have switched to door " + (switchPick + 1) + " and got a " + switchOutcome); // Check if switching would have been better if (switchOutcome == "car" && yourOutcome == "goat") { Console.WriteLine("Switching would have been better."); } else if (switchOutcome == "goat" && yourOutcome == "car") { Console.WriteLine("Not switching would have been better."); } else { Console.WriteLine("It doesn't matter if you switch or not."); } } } ```

You should switch. Initially, the probability of the car being behind each door is 1/3, but when the host opens door 3 and reveals a goat, the probability of the car being behind door 1 remains 1/3, while the probability of the car being behind door 2 becomes 2/3, since door 2 and door 3 together had a 2/3 probability of hiding the car, and door 3 has been eliminated.