Students learn how to calculate both theoretical and experimental probability by rotating through a series of work stations.

Upon completion of this lesson, students will:

• be able to calculate both experimental and theoretical probabilities
• display probabilities in both graphical and fraction form

Technological: Students must be able to:
• perform basic mouse manipulations such as point, click and drag.
• use a browser experimenting with the activities.

Students will need:

• enough stations so that each pair of students can be working at an individual station. (You may want to have multiples of each station because some stations take longer to complete than others.)
• 2 race boards and 4 race cars
• 8 dice
• 2 pieces of paper numbered 1- 12
• 10 square pieces of paper or 10 poker chips
• an opaque bag
• 15 white marbles
• 5 red marbles
• a spinner
• 3 index cards (a mole drawn on the reverse of one card)
• 2 pennies
• a deck of playing cards
• access to a browser
• paper
• pencil

1. Focus and Review
   Introduce the idea of probability through a discussion that they can relate to. Students may be familiar with winning prizes through cereal boxes or soda cans for instance. Students will be able to calculate both experimental and theoretical probabilities as well as display probabilities in both graphical and fraction form.
   
   
   
2. Teacher Input
   
   • Work through an example work station with the students.
   • Fill out the appropriate section on the with the class.
   • Explain the procedures to be followed at each station:
     Penny Flip
     1. Write whether you think the coin is more likely to land on heads or tails and why.
     2. Calculate the theoretical probability.
     3. There should be 2 pennies at the station. Each person should flip the penny and record the number of times it lands on heads and the number of times it lands on tails.
     4. Make a graph representing the results you obtained from the penny flip.
     5. Record your data on the data collection sheet.
     Spinner
     1. 1/4 of the spinner should be red, 1/4 should be green, and 1/2 should be blue.
     2. Which color do you think you are more likely to stop on and why?
     3. Calculate the theoretical probability of landing on each section.
     4. Each person should spin the spinner 50 times and record their results
     5. Make a graph representing the results you obtained from the spinner.
     6. Record your data on the data collection sheet.
     Marble Bag
     1. Place 10 of the white marbles into the opaque bag along with the 5 red marbles.
     2. Calculate the theoretical probability of drawing a red marble
     3. Draw 1 marble from the bag, record its color, and replace the marble back in the bag.
     4. Repeat step 3 until you have drawn 25 marbles.
     5. Create a graph which show the results of your experiment.
     6. Add 5 more white marbles to the bag for a total of 15 white marbles and 5 red marbles.
     7. Calculate the probability of drawing a red marble
     8. Draw 1 marble from the bag, record its color, and replace the marble back in the bag.
     9. Repeat step 8 until you have drawn 25 marbles.
     10. Create a graph which shows the results of your experiment.
     11. Record your data on the data collection sheet.
     Deck of Cards
     1. Calculate the probability of drawing a spade.
     2. Mix the deck of cards.
     3. Draw 1 card, record whether or not it is a spade, and replace it into the deck.
     4. Mix the deck of cards.
     5. Draw another card, record whether or not it is a spade, and replace the card back in the deck.
     6. Continue this process until you have drawn 20 cards
     7. Record your data on data collection sheet.
     Monte Hall Game
     1. Player 1 mixes the three cards and sets them face down so that Player 2 does not know which card is hiding the lucky mole.
     2. Player 2 chooses one of the cards.
     3. Player 1 then removes one of the losing cards.
     4. Player 2 now choses either to stay with the card he/she chose originally or switch to the card that is left.
     5. Once Player 2 has made his/her decision the group needs to record either a win or a loss result under the appropriate column on the data sheet.
     6. Player 2 then mixes the cards and sets them face down so that Player 1 does not know which card is hiding the lucky mole.
     7. Player 1 chooses one of the cards.
     8. Player 2 then removes one of the losing cards.
     9. Player 1 now choses either to stay with the card he/she chose originally or switch to the card that is left.
     10. Once Player 1 one has made his/her decision the group needs to record either a win or a loss result under the appropriate column on the data sheet.
     11. Continue this process until 24 cards have been drawn.
     12. Record your data on the data collection sheet.
     1 Step Race Car Game
     1. Player 1 is assigned the numbers 1, 2, and 3. Player 2 is assigned the numbers 4, 5, and 6.
     2. Player 1 rolls first if he/she rolls a 1, 2, or 3 he/she wins; otherwise, Player 2 rolls.
     3. If Player 2 rolls a 4, 5, or 6 he/she wins; otherwise, Player 1 rolls again.
     4. Continue this process until either Player 1 or 2 wins.
     5. Play this game at least 5 times.
     6. Record who wins each game.
     7. Calculate both the theoretical and experimental probability of each player winning.
     8. Record your data on the data collection sheet.
     9. Now try playing this game with player 1 winning on rolls of 1, 2, 3, and 4, and Player 2 winning on rolls of 5 and 6.
     10. Play this game at least 5 times.
     11. Record who wins each game.
     12. Calculate both the theoretical and experimental probability of each player winning,
     13. Record your data on the data collection sheet.
     2 Step Race Car Game
     1. Player 1 is assigned the numbers 1, 2, and 3. Player 2 is assigned the numbers 4, 5, and 6.
     2. Player 1 rolls first if he/she rolls a 1, 2, or 3 he/she wins; if not Player 2 rolls.
     3. If Player 2 rolls a 4, 5, or 6 he/she wins if not Player 1 rolls again.
     4. Continue this process until either Player 1 or 2 wins.
     5. Play this game at least 10 times.
     6. Record who wins each game.
     7. Calculate both the theoretical and experimental probability of each player winning,
     8. Record your data on the data collection sheet.
     9. Now try playing this game with Player 1 winning on rolls of 1, 2, 3, and 4, and Player 2 winning on rolls of 5 and 6.
     10. Play this game at least 10 times.
     11. Record who wins each game.
     12. Calculate both the theoretical and experimental probability of each player winning.
     13. Record your data on the data collection sheet.
     2 Dice Game
     1. Each student should number a piece of paper 2-12 and place 10 chips or paper squares on 10 numbers. The pieces of paper do not need to be placed on different numbers.
     2. Players roll the dice and the highest roll goes first.
     3. Player 1: roll the dice, calculate your sum, and record this number on your data sheet. If you have a marker on that number, remove it.
     4. Player 2: roll the dice, calculate your sum, and record the number on your data sheet. If you have a marker on that number remove it.
     5. The first player to remove all of his/her markers wins.
     6. Answer the questions located at the bottom of the data collection sheet
     7. If time permits play the game again allowing Player 2 to go first.
     8. Record your data on the data collection sheet.
   • Explain that experimental probability is the actual results gathered by doing the experiment several times.
   • Describe to the students how to calculate theoretical probability.
   • Put the students into pairs.
   
   
3. Guided Practice
   Have the students work through the stations allowing 5-10 minutes for each station.
   
   
   
4. Independent Practice
   
   • Have students rotate between the stations and complete their data collection sheet.
   • You may also want to have a computer station set up for the students to work with several probability applets that model some of the activities at the various stations.
     
     Some appropriate applets are:
     • Racing game with one die
     • Racing game with two dice
     • Adjustable Spinner Game
     • Monte Hall Game
   
   
5. Closure
   
   • Have each group share the experimental data they collected from one experiment. Ask them if the experimental probability they calculated is the same as the theoretical probability.
   • Reinforce the concepts of theoretical verses experimental probability.
   • Compile the class' data for all the experiments and compare the individual group experimental results to the collective class results. The compiled class results should be closer to the theoretical probability than most individual group's results.
   • Discuss why this is so.
   • Discuss why computers might be helpful when working with probability experiments.