- The student will use theoretical and empirical methods to estimate probabilities.
- The student will appraise the differences between the two estimates.
- The student will demonstrate an understanding of long-term relative frequencies.

Based on: Probability Topics: Probability Lab by Susan Dean, Barbara Illowsky, Ph.D.

Summary: This module presents students with a lab exercise allowing them to apply their understanding of Probability. In an experiment using M&M's candies, students will calculate and compare the theoretical and empirical probabilities of drawing particular color candies at random, with and without replacement.

- The student will use theoretical and empirical methods to estimate probabilities.
- The student will appraise the differences between the two estimates.
- The student will demonstrate an understanding of long-term relative frequencies.

Count out by color the M&M's® from a mixed-color regular bag of M&M's. Record the number of each color in the "Population" table below. Calculate the probability of each color.

Color | Quantity | Probability |
---|---|---|

Yellow (Y) | ||

Green (G) | ||

Blue (BL) | ||

Brown (B) | ||

Orange (O) | ||

Red (R) | ||

Total |

Use the information from the "Population" table to complete the theoretical probability questions.

With Replacement | Without Replacement | |
---|---|---|

Two random experiments will be undertaken. Each experiment will involve picking 2 M&M's, one at a time, 24 times. The first experiment is "With Replacement", and the second experiment is "Without Replacement. Put the M&M's from your package into a container. Do *not* look at the M&M's as you pick them.

For this first experiment, after picking the first M&M and recording its color, return it to the container before picking the second one. Record the results in the “With Replacement” column of the "Empirical Results" table below. Do this 24 times.

For this second experiment, after picking the first M&M, do *not* replace it before picking the second one. Keep track of which M&M is 1st and which is 2nd. Record the results in the “Without Replacement” column section of the "Empirical Results" table below. After you record the pick, put *both* M&M’s back in the container. Do this 24 times.

With Replacement | Without Replacement |
---|---|

( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |

( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |

( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |

( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |

( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |

( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |

( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |

( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |

( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |

( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |

( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |

( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |

Use the results in Table 3 to calculate the empirical probabilities in Table 4.

With Replacement | Without Replacement | |
---|---|---|

Use a spreadsheet program or graphing calculator to simulate a large number of trials that can be used to calculate one of the probabilities in Table 4. The random and counting functions (in Calc: RandBetween and CountIf) may be useful in programming the calculations.

Describe your simulation and probability calculation to the class and reflect on how this simulation differs from the simulation based on physically picking M&M's. How do the simulation-based probabilities compare to the theoretical probabilities (also called the expected value)? What do you think the "Law of Large Numbers" says?