Probability Topics

Do the Experiment:

Count out 40 mixed-color M&Ms which is approximately one small bag's worth. Record the number of each color inTable 3.11. Use the information from this table to completeTable 3.12. Next, put the M&Ms in a cup. The experiment is to pick two M&Ms, one at a time. Donolook 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 ofTable 3.13. Do thi 24 times. The second time through, after picking the first M&M, dono treplace it before picking the second one. Then, pick the second one. Record the results in the "Without Replacement" column section ofTable 3.14. After you record the pick, putbotM&Ms back. Do thi a total of 24 times, also. Use the data fromTable 3.14to calculate the empirical probability questions. Leave your answers in unreduced fractional form. Dono tmultiply out any fractions.

Color Quantity

Yellow (Y)

Green (G)

Blue (BL)

Brown (B)

Orange (O)

Red (R)

Table3.11Population

Count out 40 mixed-color M&Ms which is approximately one small bag's worth. Record the number of each color inTable 3.11. Use the information from this table to completeTable 3.12. Next, put the M&Ms in a cup. The experiment is to pick two M&Ms, one at a time. Donotlook 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 ofTable 3.13. Do this 24 times. The second time through, after picking the first M&M, donotreplace it before picking the second one. Then, pick the second one. Record the results in the "Without Replacement" column section ofTable 3.14. After you record the pick, putbothM&Ms back. Do this a total of 24 times, also. Use the data fromTable 3.14to calculate the empirical probability questions. Leave your answers in unreduced fractional form. Donotmultiply out any fractions.

With Replacement Without Replacement

P(2 reds)

P(R₁B₂ORB₁R₂)

P(R₁ANDG₂)

P(G₂|R₁)

P(no yellows)

P(doubles)

P(no doubles)

Table3.12Theoretical Probabilities

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.

With Replacement Without Replacement

( __ , __ ) ( __ , __ ) . ( __ , __ ) ( __ , __ )

( __ , __ ) ( __ , __ ). ( __ , __ ) ( __ , __ )

( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )

( __ , __ ) ( __ , __ ). ( __ , __ ) ( __ , __ )

( __ , __ ) ( __ , __ ). ( __ , __ ) ( __ , __ )

( __ , __ ) ( __ , __ ). ( __ , __ ) ( __ , __ )

( __ , __ ) ( __ , __ ). ( __ , __ ) ( __ , __ )

( __ , __ ) ( __ , __ ). ( __ , __ ) ( __ , __ )

( __ , __ ) ( __ , __ ). ( __ , __ ) ( __ , __ )

( __ , __ ) ( __ , __ ). ( __ , __ ) ( __ , __ )

( __ , __ ) ( __ , __ ). ( __ , __ ) ( __ , __ )

( __ , __ ) ( __ , __ ). ( __ , __ ) ( __ , __ )

Table3.13Empirical Results

With Replacement Without Replacement

P(2 reds)

P(R₁B₂ORB₁R₂)

P(R₁ANDG₂)

P(G₂|R₁)

P(no yellows)

P(doubles)

P(no doubles)

Table3.14Empirical Probabilities

Discussion Questions

1. Why are the "With Replacement" and "Without Replacement" probabilities different?
2. ConvertP(no yellows) to decimal format for both Theoretical "With Replacement" and for Empirical "With Replacement". Round to four decimal places.

a.Theoretical "With Replacement":P(no yellows) = _______

b.Empirical "With Replacement":P(no yellows) = _______

c.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 whatP(G₁ANDR₂) andP(R₁|G₂) represent. Hint: Think about the sample space for each probability.