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Probability Topics

Probability Topics

OpenStaxCollege

Probability Topics

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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&Ms® which is approximately one small bag’s worth. Record the number of each color in [link]. Use the information from this table to complete [link]. Next, put the M&Ms in a cup. The experiment is to pick two M&Ms, 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 [link]. 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 [link]. After you record the pick, put both M&Ms back. Do this a total of 24 times, also. Use the data from [link] 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(2 reds)
P(R1B2 OR B1R2)
P(R1 AND G2)
P(G2|R1)
P(no yellows)
P(doubles)
P(no doubles)
Note

G2 = green on second pick; R1 = red on first pick; B1 = brown on first pick; B2 = brown on second pick; doubles = both picks are the same colour.

Empirical Results
With Replacement Without Replacement
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
Empirical Probabilities
With Replacement Without Replacement
P(2 reds)
P(R1B2 OR B1R2)
P(R1 AND G2)
P(G2|R1)
P(no yellows)
P(doubles)
P(no doubles)

Discussion Questions

  1. Why are the “With Replacement” and “Without Replacement” probabilities different?
  2. Convert P(no yellows) to decimal format for both Theoretical “With Replacement” and for Empirical “With Replacement”. Round to four decimal places.
    1. Theoretical “With Replacement”: P(no yellows) = _______
    2. Empirical “With Replacement”: P(no yellows) = _______
    3. 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 two M&Ms to 240 times, why would empirical probability values change?
  4. Would this change (see part 3) 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(G1 AND R2) and P(R1|G2) represent. Hint: Think about the sample space for each probability.

License

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Probability Topics by OpenStaxCollege is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.