# 4.12 Lab: probability topics

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

Class time:

Names:

## 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.

## Do the experiment:

Count out 40 mixed-color M&M’s® which is approximately 1 small bag’s worth (distance learning classes using the virtual lab would want to count out 25 M&M’s®). Record the number of each color in the "Population" table. Use the information from this table to complete the theoretical probability questions. Next, put the M&M’s in a cup. The experiment is to pick 2 M&M’s, one at a time. Do not look at them as you pick them. The first time through, replace the first M&M before picking the second one. Record the results in the “With Replacement” column of the empirical table. Do this 24 times. The second time through, after picking the first M&M, do not replace it before picking the second one. Then, pick the second one. Record the results in the “Without Replacement” column section of the "Empirical Results" table. After you record the pick, put both M&M’s back. Do this a total of 24 times, also. Use the data from the "Empirical Results" table to calculate the empirical probability questions. Leave your answers in unreduced fractional form. Do not multiply out any fractions.

Population
Color Quantity
Yellow (Y)
Green (G)
Blue (BL)
Brown (B)
Orange (O)
Red (R)
Theoretical probabilities
With Replacement Without Replacement
$P\left(\text{2 reds}\right)$
$P\left({R}_{1}{B}_{2}\text{OR}{B}_{1}{R}_{2}\right)$
$P\left({R}_{1}\text{AND}{G}_{2}\right)$
$P\left({G}_{2}\text{|}{R}_{1}\right)$
$P\left(\text{no yellows}\right)$
$P\left(\text{doubles}\right)$
$P\left(\text{no doubles}\right)$
Empirical results
With Replacement Without Replacement
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
Empirical probabilities
With Replacement Without Replacement
$P\left(\text{2 reds}\right)$
$P\left({G}_{2}\text{|}{R}_{1}\right)$
$P\left(\text{no yellows}\right)$
$P\left(\text{doubles}\right)$
$P\left(\text{no doubles}\right)$

## Discussion questions

1. Why are the “With Replacement” and “Without Replacement” probabilities different?
2. Convert $\text{P(no yellows)}$ to decimal format for both Theoretical “With Replacement” and for Empirical “With Replacement”. Round to 4 decimal places.
• Theoretical “With Replacement”: $\text{P(no yellows)}=$
• Empirical “With Replacement”: $\text{P(no yellows)}=$
• Are the decimal values “close”? Did you expect them to be closer together or farther apart? Why?
3. If you increased the number of times you picked 2 M&M’s to 240 times, why would empirical probability values change?
4. Would this change (see (3) above) cause the empirical probabilities and theoretical probabilities to be closer together or farther apart? How do you know?
5. Explain the differences in what $P\left({G}_{1}\text{AND}\phantom{\rule{2pt}{0ex}}{R}_{2}\right)$ and $P\left({R}_{1}\text{|}{G}_{2}\right)$ represent. Hint: Think about the sample space for each probability.

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