Probability can be a useful tool in game design, but only when designers know how to use it. As someone who has struggled with probability in the past, frequently ponders its applications in games, and misses teaching: I’ve decided to start writing this series of articles on probability in games. To kick the series off, I’ll start with my earliest memory of frustration with probability. Looking back, echos of this early frustration have haunted many of my attempts to internalize probability over the years.

The concept of events having a certain probability or chance of occurring is not in itself a difficult one. Probabilities are typically expressed as the ratio of outcomes in which an event occurs, divided by the total number of possible outcomes. What often makes probability difficult is ensuring that you’re counting outcomes correctly: without missing, or re-counting any of them. To illustrate this difficulty, let me share the classic Monte Hall problem as it was described to me by a student teacher in grade school.

Imagine that you are on a game show (like Let’s Make A Deal) where you are told that you can keep a fantastic prize, if you can choose which of three doors it is hidden behind. You have a 1/3 chance of picking the correct door and winning the prize, because there are three doors you could possibly choose, and only one of them conceals the prize. However after you choose a door, the host opens one of the other two doors to reveal that there is no prize behind it. You are then given the option of changing your choice from the door you originally picked to the other unopened door. Are you more likely to win the prize by sticking with the door you picked in the first place, by changing your choice to the other door, or do you have the same chance of winning the prize either way?

My answer to this question was that it didn’t matter whether you switch doors or not. Since only one of the two doors conceals the prize, it’s a coin flip 1/2 chance of winning with either of the remaining doors. However my student teacher was adamant that you’re best chance at winning the prize would be to change doors after seeing the empty door revealed. This same answer was published in an issue of Parade magazine in 1990. I was outraged that this was the only acceptable answer, as were over 10,000 readers of Parade who wrote in to complain. Ever since this memorable math class, I’ve seen this solution posted on countless internet forums. And it’s often accompanied by a confused if not illogical, explanation.

The rational explanation of this solution requires that you know ahead of time that the game show host will always show you one incorrect door after you make your initial choice. Unfortunately, I don’t think I’ve ever seen or heard the problem explained with this crucial caveat. Once you know that the host will always reveal one of the two empty doors, you can start to look at your initial choice as a way of potentially limiting which non-winning door the host reveals. When your initial choice is a non-winning door, there is only one other non-winning door for the host to reveal. When this happens, you can win the prize by changing your choice of door after the reveal. Since your chance of first picking a non-winning door is 2/3, you can win 2/3 of the time by always changing doors after the revel. This is much better than the 1/3 or even 1/2 chances of blindly picking one of the available doors. However this strategy really depends on the host’s upfront honesty and predictably indifferent behavior.

So, what if the host does not explain ahead of time that they will always reveal one door? A greedy host might only give you the opportunity to change doors when you initially choose the winning door. A more philanthropic host might only give you the opportunity to change doors when you initially choose a non-winning door. You should never change your choice with a greedy host, and always change with a philanthropic judge. I suspect that many people’s frustration with the Monte Hall problem’s solution, comes from their skepticism about the human nature of game show hosts.

        Host Type:     | Indifferent Opener | Greedy Opener | Philanthropic Opener
        ---------------+--------------------+---------------+---------------------
        Best Strategy: |    Always Change   |  Never Change |    Always Change
        ---------------+--------------------+---------------+---------------------
        Chance to Win: |        2/3         |      1/3      |         3/3

This problem illustrates one of many difficulties in counting outcomes. Something as subtle as the motivation of a game show host can inform our strategies for changing doors, and result in different numbers of winning outcomes. We could extend our solution above by speculating on the probabilities that a random game show host falls into each category. However we might miss other categories of host motivations, For example, a game show host might try to maximize the perception that they are not acting greedily, while in fact trying to minimize the number of prizes they award across several game shows.

Hopefully this (and future articles in the series) leaves you with more questions than answers. I encourage you to further explore those questions and comment on any that catch your fancy. Some of those questions could lead to interesting followup comments, and others might be worth visiting in a full article format.




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Tuesday, January 24th, 2012