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# Probability Topics: M&M Lab

Module by: Alison Snieckus. E-mail the author

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.

## Student Learning Outcomes:

• 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.

## Theoretical Probabilities

### The population

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.

Table 1: Population
Color Quantity Probability
Yellow (Y)
Green (G)
Blue (BL)
Brown (B)
Orange (O)
Red (R)
Total

### Theoretical probability questions

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

Table 2: Theoretical Probabilities
Note: G 2G 2 = green on second pick; R1R1 = red on first pick; B1B1 = brown on first pick; B2B2 = brown on second pick; doubles = both picks are the same colour.
With Replacement Without Replacement
P(2 reds)P(2 reds)
P ( R1 B2 OR B1 R2 )P(R1B2 OR B1R2)
P ( R1 AND G2 )P(R1 AND G2)
P ( G2 | R1 )P(G2 | R1)
P(no yellows)P(no yellows)
P(doubles)P(doubles)
P(no doubles)P(no doubles)

## Empirical Probabilities

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.

### With Replacement

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.

### Without Replacement

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.

Table 3: Empirical Results
With Replacement Without Replacement
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )

### Empirical probability questions

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

Table 4: Empirical Probabilities
With Replacement Without Replacement
P(2 reds)P(2 reds)
P ( R1 B2 OR B1 R2 )P(R1B2 OR B1R2)
P ( R1 AND G2 )P(R1 AND G2)
P ( G2 | R1 )P(G2 | R1)
P(no yellows)P(no yellows)
P(doubles)P(doubles)
P(no doubles)P(no doubles)

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?

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