
1. An urn contains 3 Blue, 4 Green, and 5 Yellow marbles, labeled from 1 to
12. B1...B3,G4, ...G7, Y8,...Y12. One marble is randomly selected.
2. List the sample space as {B,G,Y} and their corresponding probabilities:
P(B)=____; P(G)=_____; P(Y)=______
3. Now we run an experiment on Excel to check the above probabilities.

Method 1: 
1) In A1, type in "=randbetween(1,12)". This gives the number of the
marble
2) in B1, type in"=if(A1<4, "B", if(A1>7,"Y","G")) ". This changed the
number to the color.
3) Copy A1:B1 to A2:B200. This gives a sample for 200 experiments.
4) Use B1:B100 as the sample, construct the frequency table by using pivot
table.

Method 2:   you can type in B, B,B, G,G,G,G,Y,Y,Y,Y,Y in A1:A12, then in B1

=index($A$1:$A$12, randbetween(1,12) ,1)

and copy B1 to B2:B200 . This will also generate 200 samples. 

4. Use both method to get a frequency table and compare with the true probabilities in 2), and discuss how much
differences you see.

### 2. Sample Space and Corresponding Probabilities

Given the distribution of marbles:
- Blue: 3 marbles (B1, B2, B3)
- Green: 4 marbles (G4, G5, G6, G7)
- Yellow: 5 marbles (Y8, Y9, Y10, Y11, Y12)

#### Sample Space
\[
S = \{B, G, Y\}
\]

#### Corresponding Probabilities
Since one marble is selected randomly, the probability of selecting each color is proportional to the number of marbles of that color.

- **Total number of marbles** = 3 (Blue) + 4 (Green) + 5 (Yellow) = 12

Thus:
- $ P(B) = \frac{3}{12} = 0.25 $
- $ P(G) = \frac{4}{12} = 0.333 $
- $ P(Y) = \frac{5}{12} = 0.417 $

### 3. Experiment Implementation in Excel

#### **Method 1**: Using `randbetween()` and `if()` functions

1. **A1**: Type `=randbetween(1,12)` to randomly select a marble numbered from 1 to 12.
2. **B1**: Type `=if(A1<4,"B",if(A1>7,"Y","G"))` to convert the random number into the corresponding color (Blue for numbers 1–3, Green for numbers 4–7, and Yellow for numbers 8–12).
3. Copy **A1:B1** down to **A2:B200** to simulate 200 random experiments.

- Now use the **Pivot Table** feature in Excel to create a frequency distribution of colors in the range **B1:B200**. The pivot table will show how often Blue, Green, and Yellow appear.

#### **Method 2**: Using `index()` and `randbetween()` functions

1. In column **A1:A12**, manually type the color of each marble:
    - A1:A3 = "B" (Blue)
    - A4:A7 = "G" (Green)
    - A8:A12 = "Y" (Yellow)
  
2. In **B1**, type `=index($A$1:$A$12, randbetween(1,12), 1)` to randomly select one color based on the list of marbles.
3. Copy **B1** down to **B2:B200** to simulate 200 random experiments.

- Again, use the **Pivot Table** feature in Excel to construct the frequency table for the color distribution based on this experiment.

### 4. Frequency Table and Comparison with True Probabilities

Once you've generated the frequency tables for both Method 1 and Method 2, you'll have observed relative frequencies of Blue, Green, and Yellow marbles from your 200 trials. Compare these experimental frequencies with the theoretical probabilities calculated in step 2.

#### Discussion
The results from both methods should approximate the true probabilities:
- $ P(B) = 0.25 $
- $ P(G) = 0.333 $
- $ P(Y) = 0.417 $

However, due to the randomness and limited number of trials (200), there might be some variation between the observed frequencies and the theoretical probabilities. If you increase the number of experiments, say to 1000 trials or more, the experimental probabilities will likely converge closer to the theoretical values due to the Law of Large Numbers.

### Do you want help with how to set up the pivot table in Excel? Or further discussion on the probability?

#### 5 Related Questions:
1. How does increasing the number of trials affect the accuracy of experimental probabilities?
2. What is the Law of Large Numbers, and why does it apply to these experiments?
3. How would the results differ if you only performed 50 experiments instead of 200?
4. Can you modify Method 2 to simulate an experiment with different numbers of marbles for each color?
5. How can you automate the frequency count without using a pivot table?

**Tip**: To ensure accuracy, always run more trials to reduce the impact of random fluctuations in your experimental data.

1. An urn contains 3 Blue, 4 Green, and 5 Yellow marbles, labeled from 1 to 12. B1...B3, G4, ...G7, Y8,...Y12. One marble is randomly selected. List the sample space as {B,G,Y} and their corresponding probabilities: P(B)=____; P(G)=_____; P(Y)=______. 2. Now we run an experiment on Excel to check the above probabilities using two different methods and compare the results.

Explore how to calculate the probability of selecting Blue, Green, or Yellow marbles from an urn using Excel simulations. This problem introduces the concepts of probability, random sampling, and the Law of Large Numbers. Suitable for Grades 9-12.

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