Probability Topics
Probability Topics
OpenStaxCollege
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&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.
Color | Quantity |
---|---|
Yellow (Y) | |
Green (G) | |
Blue (BL) | |
Brown (B) | |
Orange (O) | |
Red (R) |
With Replacement | Without Replacement | |
---|---|---|
P(2 reds) | ||
P(R_{1}B_{2} OR B_{1}R_{2}) | ||
P(R_{1} AND G_{2}) | ||
P(G_{2}|R_{1}) | ||
P(no yellows) | ||
P(doubles) | ||
P(no doubles) |
G_{2} = green on second pick; R_{1} = red on first pick; B_{1} = brown on first pick; B_{2} = brown on second pick; doubles = both picks are the same colour.
With Replacement | Without Replacement |
---|---|
( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |
( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |
( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |
( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |
( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |
( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |
( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |
( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |
( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |
( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |
( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |
( __ , __ ) ( __ , __ ) | ( __ , __ ) ( __ , __ ) |
With Replacement | Without Replacement | |
---|---|---|
P(2 reds) | ||
P(R_{1}B_{2} OR B_{1}R_{2}) | ||
P(R_{1} AND G_{2}) | ||
P(G_{2}|R_{1}) | ||
P(no yellows) | ||
P(doubles) | ||
P(no doubles) |
Discussion Questions
- Why are the “With Replacement” and “Without Replacement” probabilities different?
- Convert P(no yellows) to decimal format for both Theoretical “With Replacement” and for Empirical “With Replacement”. Round to four decimal places.
- Theoretical “With Replacement”: P(no yellows) = _______
- Empirical “With Replacement”: P(no yellows) = _______
- Are the decimal values “close”? Did you expect them to be closer together or farther apart? Why?
- If you increased the number of times you picked two M&Ms to 240 times, why would empirical probability values change?
- Would this change (see part 3) cause the empirical probabilities and theoretical probabilities to be closer together or farther apart? How do you know?
- Explain the differences in what P(G_{1} AND R_{2}) and P(R_{1}|G_{2}) represent. Hint: Think about the sample space for each probability.