Probability Topics
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 ExperimentCount out 40 mixed-color candies, which is approximately one small bag’s worth. Record the number of each color in Table 3.12. Use the information from this table to complete Table 3.13. Next, put the candies in a cup. The experiment is to pick two candies, one at a time. Do not look into the cup as you pick them. The first time through, replace the first candy before picking the second one. Record the results in the With Replacement column of Table 3.14. Do this 24 times. The second time through, after picking the first candy, do not replace it before picking the second one. Then, pick the second one. Record the results in the Without Replacement column section of Table 3.15. After you record the pick, put both candies back. Do this a total of 24 times, also. Use the data from Table 3.15 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) |
Note
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 color.
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 candies to 240 times, why would empirical probability values change?
- Would this change (see Question 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.