BOOMbox at Home: Probability and Strategy
June 11, 2021
This week, we’re diving into probability and the strategies people use to win games.
Try playing this matchstick game. How quickly can you figure out the secret?
After a few rounds, you might find yourself feeling a bit stuck. If you can’t figure out how to win, you aren’t alone. Depending on your perspective, this game is either impossible or a guaranteed success. The second player, in this case, the computer, has the advantage. No matter what decisions you make, the second player can always win. You can prove this by mapping out the different choices you could make. For some extra fun, try playing with a family member and “generously” offer to let them go first.
Games can be based on chance, skill, or a mixture of the two. Most games are somewhere in between--a skilled player can make decisions that increase their chances of winning, but luck can make things easier or more difficult. In a game that is entirely skill-based, the outcome is just a result of which player makes the fewest mistakes. Sometimes, this gives you situations where one player can guarantee a win. In others, it means that either player can tie the game or keep it going forever. Think of tic-tac-toe. There are a limited number of possible choices, and if you know all of the ways to win, you can always find ways to block your opponent. Can you make a list of rules, also known as an algorithm, to determine the best move to make in any situation?
A more complex example is the L-game. In this game, each player takes turns moving their L-shaped piece, and, if they choose, moving one of the two neutral pieces. The goal is to create a situation where your opponent has no available moves. Although it only takes a few minutes to learn the rules of this game, the constantly changing environment makes it challenging to master. If both players always make the best possible move, the game can last forever, but most of the time, one or both of the players will make moves that are good but not perfect, giving the other player a chance to win. Learn more about the L-Game or try playing yourself.
To Swap or Not to Swap
There are three doors. One door holds a fantastic prize. The other two have nothing. Choose a door. After you choose, I open one of the other doors to show that it is empty. Then, I give you the choice to either stick with your original choice or swap to the other unopened door. Once you make your final decision, you’ll receive whatever is behind your door. Should you swap or stick with your original choice?
This question is called the Monty Hall problem, named after a game show host, and it’s been stumping people for decades! Most people assume that, since there are three doors and one prize, each door has a ⅓ chance of having the prize. So it shouldn’t matter whether you swap, or should it? Let’s find out! Try out this simulation to see which strategy--staying or switching--wins most often.
Did you find that switching was usually the better choice? Weird, right? There are a few different ways to think about this. One approach is: when you initially choose a door, you’re creating two “teams,” your team, with one door, has one chance at winning, and the other team, with two doors, has two chances at winning. The host knows what is behind each door, or the secret identity of each team member. They will never show you which door is secretly the prize.
When you have the option to swap, you’re choosing whether to stick with the team with one chance or move to the team with two chances. Even though you now know one of the wrong answers, that doesn't change the fact that that team still counts for TWO chances rather than one. The odds have not changed, but the single closed door on that team now bears all of the team’s chances. Watch an explanation of the Monty Hall problem.
For now, let’s think about another game show: Deal or No Deal. In Deal or No Deal, you start by picking 1 of 26 cases. Each case contains a different amount of money, from $0.01 to $1,000,000. Throughout the game, you, without knowing what is any of the cases, choose some cases to open. Because each case has a different amount of money, you know that any amount you find in another case can’t be the amount in your case. Throughout the game, you are given offers to sell your case.
If you sell, you get the amount offered and lose the chance to get the amount in your case. At the end of the game, if you are still playing when there is only one case left, you can choose to keep your case or swap for the other case. You take home the amount in whichever case you choose. Do you think this game works the same as the door game or differently? Should you switch? Or should you sell your case before you even get that far? How do the amounts in the other cases change your decision making?
Hit or Stay
Blackjack is a popular card game often played in casinos. Casino games almost always have odds in favor of the casino--that’s how they stay in business. However, Blackjack has one of the smallest margins. A skilled player can have almost even odds, and some of the basic fundamentals of strategy are easy to learn. So, Blackjack can be a simple and fun way to experiment with probability and strategy at home--no gambling required!
Here’s how it works: in Blackjack, players compete to reach a total of twenty-one without going over. Number cards are worth their numerical value, and face cards (jack, queen, king) are each worth 10. An ace can count as either 1 or 11, whichever is most beneficial to the player. Each player starts with one card face down and one card face up. After looking at your cards, you can choose to hit (take another card) or stay (refuse another card). Once all players have chosen to stay, the face down cards are revealed and totals are tallied.
Because you don’t know what cards your opponent has, you must make inferences about their total in order to compare it to yours. If they have a 7 showing, their total could be anything from 8 to 18. Sometimes, the correct choice is obvious. For example, if you have a total of 8, without doing any math, you can safely say that you should probably take another card, since 8 isn’t very close to 21. Likewise, if you have a total of 18, you can safely say that you probably shouldn’t take another card.
But what if you’re somewhere in between? Say for example that you have a total of 15. On the one hand, that feels like it’s probably too low to win, but on the other hand, if you take another card, you might bust. What should you do? Play a few games yourself at home to get a sense of what works. Then, take a look at this Blackjack guide to see, based on the math, if your strategy is the best approach.
Let’s play a game. We’ll each choose a number from 1 to 6 and then we’ll roll a six-sided die 20 times. Whoever’s number comes up the most often wins. What number will you pick? Well, it shouldn’t really matter. Assuming the die is fair, each number has an equal chance of being rolled. It’s just a game of chance. But what if we change the rules a bit? We’ll each pick a number from 2-12, and then we’ll roll two six-sided dice. Should you still pick just any number, or is there more strategy involved this time? Find out by doing the experiment yourself. You may already have dice around your house, but if not, you can always make your own. Roll two six-sided dice 20 times. For each roll, write down both of the individual numbers as well as the total, for example: roll 1 result: 2 +3 = 5. What patterns do you notice?
Okay, okay, just rolling dice probably isn’t a very interesting game. But, the information you learn from this sort of experiment can help you devise a strategy for more complex games. Consider for example Settlers of Catan. In this board game, players compete to build empires and collect victory points. There are lots of ways to score points, but they all come back to resources. Players use combinations of 5 different resources (wood, brick, wheat, sheep, and ore) to build, make upgrades, and purchase bonuses.
Resource distribution is based on rolling the dice. Each turn, a player rolls two dice to determine which tiles produce resources. Players with settlements touching those tiles collect the relevant resources. So, no matter your long-term strategy, it’s really helpful to have an idea of which tiles will produce the most resources, or, what numbers are rolled the most often. And, after completing this experiment, we bet you can probably guess why Catan has special rules for 7!
Person of the Week
Concha Gómez is an Italian and Cuban-American mathematician and advocate for diversity in math. Despite a lack of support from her family, she obtained a BA and PhD in mathematics from University of California. While working towards her doctorate, she cofounded “The Noetherian Ring,” a group for women in math. Although she had to adjust the course of her career due to the symptoms of an autoimmune disorder, she continues to teach and mentor. She previously led Wisconsin Emerging Scholars (EWS), an emerging scholars program at the University of Wisconsin that was designed to provide additional support for minority students. She currently works with Spectra, an organization for LGBTQ+ mathematicians, and teaches at Diablo Valley College.
Written by Eli, Michelle, and Veena.