Replacement and non-replacement probabilities with dice and ducks (Investigation)

Objectives

• To explore probability in an engaging activity
• To practice with relative frequencies, replacement vs. non-replacement, independent vs. dependent events, conditional probabilities and counting techniques

Materials

• Large bowl or bucket approximately 20 inches in diameter
• 12 Small rubber ducks
• Permanent marker
• Water
• Two dice
• Pen
• Paper

Procedure

Work in pairs or in small groups.

SET UP

1. Write numbers 1-12 on the bottom of the ducks using the permanent marker.
2. Fill up the bowl/bucket with water.
3. Make sure the marker on the bottom of the rubber ducks is dry, then place the rubber ducks into the water so that they float. 

HOW TO PLAY

1. First, roll both of the dice. Add together the numbers you rolled.
2. Then, pick a duck from the water and look at the number marked on the bottom.
3. Record both numbers.

Pre- Investigation Questions

Before beginning the investigation answer the following questions. 

1. If you were to replace the duck you chose after rolling the dice what would be the probability of:
   • Rolling a sum of 6 on the dice and picking a duck labeled 6.
   • Rolling at least a sum of 4 on the dice and picking a duck labeled 8.
   • Rolling an even sum on the dice and picking a duck with an even number.
   • Rolling an odd sum on the dice and picking a duck with an even number.
   • Picking a duck labeled with the sum rolled on the dice.
   • Not rolling a sum of 6 and not picking a duck labeled with a 4.
2. If you were to draw the ducks with replacement how many different ways can you draw the numbers on the bottom of the ducks?
3. If the duck is drawn with replacement after the dice are rolled how many different combinations of the duck number and the sum of the dice are there?
4. If you were to roll the dice and pick a duck with replacement twice in a row what is the probability of:
   • Not picking a duck with a 6 on it both times.
   • Picking a duck with a 5 on it and a duck with a 10 on it.
   • Picking a duck with a 7 on it and not a 4.
   • Rolling an odd sum given that a 4 is rolled on one of the dice.
   • Picking a duck with a number equal to or greater than 7.
   • Rolling an odd sum on the dice both times.
5. If the duck is drawn without replacement how many different ways can you draw the numbers on the bottom of the ducks? How is this different than the amount of ways you could draw these numbers when the ducks are replaced?
6. If you were to roll the dice and pick a duck without replacement twice what is the probability of:
   • Not picking a duck with a 6 on it both times.
   • Picking a duck with a 5 on it and a duck with a 10 on it.
   • Picking a duck with a 7 on it and not a 4.
   • Rolling an odd sum given that a 4 is rolled on one of the dice.
   • Picking a duck with a number equal to or greater than 7.
   • Rolling an odd sum on the dice both times.

Investigation

1. First complete 15 trials with replacement of the duck. One full trial entails rolling the dice, finding the sum of the numbers, and then picking a duck. Be sure to record your results for each trial on a table similar to the one below and label it “Trials With Replacement.”

   Sum of Rolled Dice

   1.  
   2.  
   3.  
   4.  
   5.  
   6.  
   7.  
   8.  
   9.  
   10.  
   11.  
   12.  
   13.  
   14.  
   15.  

   Number on Duck

   1.  
   2.  
   3.  
   4.  
   5.  
   6.  
   7.  
   8.  
   9.  
   10.  
   11.  
   12.  
   13.  
   14.  
   15.  
2. Next, complete 15 trials without replacement of the duck. One full trial entails rolling the dice, finding the sum of the numbers, and then picking a duck which you will not replace. Be sure to record your results for each trial on a table similar to the one used previously but title it “Trials Without Replacement.”

Post Investigation Questions

1. In the trials you performed with replacement is the act of choosing a duck dependent or independent of previous trials? What about the act of rolling the dice?
2. In the trials you performed without replacement is the act of choosing a duck dependent or independent of previous trials? What about the act of rolling the dice?
3. Use the information on your chart labeled “Trials With Replacement” to find the following observed probabilities:
   • Rolling a sum of 6 on the dice and picking a duck labeled 6.
   • Rolling at least a sum of 4 on the dice and picking a duck labeled 8.
   • Rolling an even sum on the dice and picking a duck with an even number.
   • Rolling an odd sum on the dice and picking a duck with an even number.
   • Picking a duck labeled with the sum rolled on the dice.
   • Not rolling a sum of 6 and not picking a duck labeled with a 4.
   • Not picking a duck with a 6 on it two times in a row.
   • Picking a duck with a 5 on it and then a duck with a 10 on it.
   • Picking a duck with a 7 on it and then a duck without a 4 on it.
   • Rolling an odd sum given that a 4 is rolled on one of the dice.
   • Picking a duck with a number equal to or greater than 7.
   • Rolling an odd sum on the dice two times in a row.
4. Use the information on your chart labeled “Trials Without Replacement” to find the following observed probabilities:
   • Not picking a duck with a 6 on it two times in a row.
   • Picking a duck with a 5 on it and then a duck with a 10 on it.
   • Picking a duck with a 7 on it and then a duck without a 4.
   • Rolling an odd sum given that a 4 is rolled on one of the dice.
   • Picking a duck with a number equal to or greater than 7.
   • Rolling an odd sum on the dice two times in a row.
5. Were the probabilities you calculated from the data you collected close to your predictions? Why do you think that is?
6. Either combine your data with a friend’s or repeat the trials so that you have more data. Recalculate the probabilities that were farthest from your predictions. Did the addition of data affect them? Why do you think that is?