Pick a Card, Any Card

Objective

In this experiment you will test if the probability of drawing a particular card from a deck depends upon the number of that type of card in the deck.

Introduction

Each time you draw a card in a card game you have a certain chance at getting the card you need to beat your opponent and win the game. Consider the game Go-Fish, being played with a regular deck of playing cards. The object of the game is to win the most four of a kind sets by asking your opponent for matching cards or by drawing matching cards from the deck. At the end of the game, the player with the most sets of matched cards wins. If you want to win the game, you need to increase your chances of getting matching cards, but how?

By understanding how chance is related to math, you can learn to play with a winning strategy. For example, what if you have three kings and one queen in your hand and it is your turn to ask for a card, which one should you ask your opponent for? At first you might think that because you have more kings, you should ask for those, but it is actually better to ask for the queen! Why? Because you have a better chance of getting it!

Here is how you can figure it out: There are only four of each kind of card in the deck, so there are four kings and four queens total. If you have three kings in your hand, there is only one king left. Since only one queen is in your hand, there are three queens left. You have one chance to get a king, but three chances to get a queen out of the remaining cards. That is why you have a better chance of getting the queen than the king if you ask for it.

At first this sounds very confusing, but the more you try it you will see how it works. This strategy is based on something called probability which is how mathematicians study how likely an event is. There are many events that can be described by probability and math, especially in games we like to play. The chance that you will draw a certain type of card in a game of Go-Fish, the chance that you will roll a six in Chutes and Ladders or the chance that you will spin green when playing Twister are all probabilites.

The nice thing about a probability is that you can measure it by counting and using some very basic math, like addition and division. In the example above, I knew that there were four kings in a deck of cards because I can count them. I can use addition and subtraction to know how many I have left. In this experiment, you will measure the probability of drawing specific types of cards from a deck. You will choose which cards to try for, and then measure your success at drawing the card. Which cards will be the easiest to draw? Which are the most difficult? Will your chances of drawing the card be related to how many of that card are in the deck? How can probability help you choose the right strategy?

Terms, Concepts and Questions to Start Background Research

To do this type of experiment you should know what the following terms mean. Have an adult help you search the Internet, or take you to your local library to find out more!

• probability
• chance
• strategy
• likelihood

Bibliography

• Burns, Marilyn. 1994. "4 Great Math Games: Favorite Activities From Marilyn Burns to Try in Your Classroom." Instructor, Scholastic Books, Inc. [2/25/06]
  http://teacher.scholastic.com/lessonrepro/lessonplans/grmagam.htm
• McLeod, John. 2005. "Classified Index of Card Games." Pagat.com [2/25/06]
  http://www.pagat.com/class/

Materials and Equipment

• a deck of playing cards
• notebook
• pencil
• calculator

Experimental Procedure

1. Prepare the deck of cards for your experiment. Count the cards to make sure the deck is complete (each deck should have 52 cards total). Remember to take out the jokers! Shuffle the deck three times and set aside.
2. Prepare a data table for your data. The table should include space to write all of your observations, including the name of the type of card, the number of the type of card in the deck, the number of cards drawn for each trial and space to add together and average your data. Here is an example of a data table for this experiment:

3. Choose your first type of card and write it in the first column of your data table. Count how many of that type of card there are in the deck and write this number in your table.
4. Draw cards from the top of the deck and flip them over one at a time, counting as you go. When you get to the type of card you are looking for, stop and write down the number of cards you have drawn in your table. This will be your first trial.
5. Shuffle the cards and repeat step 4 nine more times to get a total of ten trials for the first type of card.
6. Repeat steps 3–5 for each type of card you would like to test (that is, for each column in your data table).
7. Now you will want to tally up your data by adding together the number of cards drawn for the ten trials in each column. Write your answer in the "TOTALS" row. Are the numbers similar or different?
8. Next, you will want to calculate an average for each experiment. The average is a way to combine the results of all of your trials into one number, which will be useful for graphing and understanding the results of your experiment. Do this by dividing the number in each "TOTALS" box by ten, and writing the answer below in the "AVERAGES" box of the data table.
9. Now you can analyze your data by making a few graphs:
   • You can use a bar graph to show the average number of cards that were drawn for each specific type of card you wanted. On the left side of the graph (Y-axis) you will put a scale for the average number of cards drawn, and on the bottom of the graph (X-axis) you will put your bars and labels. Use one bar for each type of card and draw the bar up to the corresponding number on the left (Y-axis) of the graph. Which cards were the most difficult to draw? The easiest to draw? Were there any similarities or differences between different types of cards and the likelihood that you could draw them?
   • You can also make a line graph showing how the number of cards drawn compares to the number of that type of card in the deck. On the left side of the graph (Y-axis) you will put a scale for the average number of cards drawn, and on the bottom of the graph (X-axis) you will put a scale representing the number of each type of card in the deck. Did you need to draw more or less cards to choose cards that were rare compared to cards that were common?
10. Interpret your results. Did the number of each type of card present in the deck change the number number of cards it took to pick the card you wanted? Which cards were most likely to be drawn? The least likely? Did the probability of choosing a specific type of card change depending upon it's representation in the deck? Make a conclusion.

Variations

• A more advanced way of showing the results of your experiment would be to make histograms, which are a type of graph to show distributions. They are especially useful for visualizing probabilities. Try making a separate histogram for each type of card you tested. Do this by graphing the number of cards drawn for each trial spearately in a bar graph. When all of the bars are lined up next to each other, what does the overall shape of the distribution look like?
• The probability of drawing a particular type of card also depends upon the number of cards drawn each time. Try another experiment to see how your chances of drawing a particular card change as you draw more cards each time. Try drawing samples of 3 cards, 5 cards, or 7 cards. Do your chances improve as more cards are taken?
• Probabilities also change as cards are removed from or added to the deck. Try the experiment again, but this time remove cards from the deck before your experiment. Try using two decks of cards combined together. Does your data change? Why or why not? Try removing select cards from the deck, like taking out half of the red or black cards, before doing the experiment. Will this change your chances? What if you left the Jokers in the deck? How would this change your results?
• Probabilities can change your strategies for playing a card game. Can you do an experiment to show how probabilities can help you choose cards when playing Go-Fish? What about other popular cards games like War, Memory, or Solitaire? Can you develop rules for a winning strategy? Can you invent your own card game based on probabilities?

Source: http://www.sciencebuddies.org

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