Saturday, April 10, 2010

A 21 Setup to Teach Probabilistic Decision Making

I want to expand on our discussion of games of chance as teaching tools. I picked the classic casino game of 21 because it is relatively simple to learn. I invented a simple variation and tested it successfully with my daughter. Hence, I assume this should work with other kids as well. The rest of this article explains the setup.

Rationale for My Version of the Game of 21
This is a modified version of 21. It removes some of the complexities that make the game inappropriate for teaching young kids about probabilistic decision making. Students say "hit"to be dealt additional cards until the probability of going bust (i.e. getting more than 21) is unacceptable. At this point, they say "stay." The point of this game is not to teach betting but to make the computation of probabilities a bit more fun than traditional classroom teaching.

Playing this game should teach:

  1. Computing probabilities on a discrete space.
  2. Computing conditional probabilities when there the probability space changes.
  3. Determining if a bet is unfavorable, roughly fair, or favorable.
  4. Making decisions based on probabilities.

As you can probably surmise from the above, the goals of my game are a bit ambitious. However, this game could give you a start to help your little one learn tools that could prove very valuable later in academia and life.

Step 1
Get a big piece of construction paper -- the kind used for school projects and presentations. Arrange a deck of cards along columns on the construction. The leftmost column should have the twos. The next column over should hold the threes and so on. Instead of using a deck of cards, you could simply draw the cards on the construction paper. You should have 13 columns total. I will call this the "board" throughout the rest of this article.

The purpose of the board is to keep track of the cards that have been played. To this end, use a spare deck of cards facing down to cover all the locations on the board corresponding to cards that have either being played or are "in play." This makes it easy to visually determine how many and which cards remain on the deck.

Step 2
Explain the rules of this version of 21:

  1. At the beginning of a hand, each player is given two cards.
  2. Each player gets only one card at a time after the initial two. Additional cards are only handed out when it is the player's turn and the player asks to be hit.
  3. The game continues until there are no cards left in the deck or there are not enough cards to start a new hand.
  4. There is a common pile of chocolate chips for betting. All players draw chips from the same pile.
  5. The winner of the game is the one who ends with the biggest pile of chocolate chips.
  6. A player should be dealt as many cards as desired until either he or she goes bust or decides to "stay."

Step 3
Explain how to play a hand.

  1. Start by getting two cards.
  2. Continue saying "hit me" (i.e. asking for another card) until you think you will go bust.
  3. Say "stay" when you are done with your hand.
  4. Choose to hit or stay based on the probability of going bust by picking one more card. This step is crucial. One of the key reasons for the this game is to teach how to compute conditional probabilities (i.e. probability under variable change).
  5. If your total exceeds 21, you are out of this hand.

Step 4
Explain the rules for betting and scoring in this game.

  1. Points are earned or lost in each hand.
  2. A bet is from 0 to 4 chips.
  3. The person who comes closest but not over 21 wins the hand and takes all the chips bet in the round.
  4. If two or more players tie, they split the pile. If the pile does not split evenly, remove the smallest number of chips so it splits evenly. Add the removed chips to the next hand's pile.

Step 5
Teach some betting guidelines.

  1. Bet 0 chips if there is a high probability of losing the hand.
  2. Bet 1 or 2 chips if there is low probability of winning the hand.
  3. Bet 2 chips if the probability of winning is high.
  4. Bet 4 chips if you are very sure to win.

A Few Additional Details
The game could be played with different degrees of sophistication. For instance, students could be asked to base the size of their bets on the probability of winning the hand. However, this requires computational skills beyond the skills of most seven year olds. A better way to place bets is based on intuition.

This game probably would be a lot of fun to play with a group of kids. I explained the setup to my daughter's teacher, and she thought it would be fun to play in the classroom since they are learning the basics of probability.

Have fun,


Wednesday, April 7, 2010

Vegas at Home

I just finished three initiation reports -- I follow stocks for a living -- and needed a diversion. So, I decided to teach Paulina how to play cards. It occurred to me earlier today that card games are a great vehicle to teach basic probability concepts. Conditional probability, in particular, arises naturally. We tried it, and Paulina had a lot of fun.

Here is what we did. I first taught her the various suits and the cards in each suit. That took very little time. I then asked her the following questions:
  1. Before dealing any cards, what is the chance of dealing a heart?
  2. Assuming that a heart is the only card that has been dealt, what is the probability of dealing another heart?
  3. Assuming that a heart is the only card that has been dealt, what is the probability of dealing a spade?
  4. Assuming that a heart has been dealt, what is a better bet for the next card? A heart or a diamond?
Shuffle the deck. Deal two cards for you and two for your child. Forget about the dealer's cards. This just complicates the setup. Now is when things get educational. Let the child be the first player. Ask the kid "hit" or "stay." He or she is going to look at you funny. Explain that you say "hit" if you want more cards and '"stay" if you are done. Remind the child that an "A" works as a 1 or an 11. Remind the child going over 21 gets you busted. Tell the kid to look at all the cards that have been dealt and ask for the probability of going bust. Chances are the kid won't have a clue. In fact, Paulina had no idea where to start. However, she got it quickly once I showed her the thought process. If your child has been paying attention, he or she will get it too. Here is an example. Let's say that you dealt:
  • a 10 of hearts and a 5 of clubs for your child
  • a 2 of diamonds and 5 of diamonds for you
  • a 10 of hearts and a card facing down
You can compute the probability of your child going bust as follows:
  • To bust, you must deal a card with a value of 7 or higher.
  • Given what has already being dealt, the following cards will get you busted: 4 sevens, 4 eights, 4 nines, 4 tens, 4 jacks, 4 queens, and 4 kings
This is a total of 28 cards out of 48 cards that have yet to be dealt. This means that the probability of going bust is 28 out of 48 or 7/12. Your child should know that this is more than 1 out of every two cards. This is not a lot over 1/2, but one is more likely than not to go bust at this point by taking one more card.

You could continue playing the game. As you work through the deck, you can ask your child to remember what has been dealt and to decide on every play whether or not it is a good idea to take another card.

Remember that this is not Vegas. You can count cards here. Lay out in front of you the cards that have been used so your kid can tell what is left in the deck. Don't be too serious. That's not the point. Have fun. Raise the stakes by using chocolate chips to bet. Let me repeat this. HAVE FUN. EAT CHOCOLATE. COMPUTE. THINK. If you make this too serious, you will fail.

There are a million questions you could ask. For example, you could ask the probability of getting a 21 when dealt the first two cards from a deck. However, you should just play and ask the kid to make educated guesses about whether or not to bet. Once again. Raise the stakes by using chocolate. Trust me. Chocolate works when teaching math.

Have fun,