No, when you have two doors, the probability of winning is not 50% if you make another choice. Let's take a closer look at why this is the case.

In the Monty Hall problem, the scenario starts with three doors: one hiding a valuable prize (let's say a car) and the other two hiding goats. Your goal is to choose the door with the car behind it.

Initially, the probability of choosing the correct door by random chance is 1/3 (33.3%). Now, let's say you randomly select Door 1.

At this point, the host, who knows what's behind each door, opens one of the other two doors that you didn't choose, revealing a goat behind it (let's say it's Door 3). Here's where the key factor comes into play:

By revealing a goat, the host is providing you with valuable information. The probability distribution has now changed. The car is either behind Door 1 (your initial choice) or Door 2 (the remaining unopened door).

If you stick with your initial choice (Door 1), your probability of winning remains at 1/3. However, if you switch your choice to the other unopened door (Door 2), your chances of winning increase.

To understand why switching is advantageous, consider the two possible scenarios:

1.You initially chose the door with the car (Door 1). If you switch, you will lose because the car is no longer behind that door.

2.You initially chose a door with a goat (Door 1). If you switch, you will win because the car must be behind the remaining unopened door (Door 2).

In this scenario, switching doors yields a 2/3 (66.7%) probability of winning, while sticking with your initial choice only has a 1/3 (33.3%) chance of winning.

This counterintuitive result can be explained by the fact that the host's action of revealing a goat provides new information that affects the probability distribution. By switching doors, you are essentially leveraging that information and increasing your chances of selecting the door with the car behind it.

So, in the Monty Hall problem, switching doors gives you a higher probability of winning (66.7%), while sticking with your initial choice would only give you a 33.3% chance of winning.
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