Question

Error Analysis  The sample space of picking a 2 character password using the letters Upper G​, Upper O​, Upper P​, Upper R​, Upper Y​, Upper B​, where double letters are not allowed is shown. Your teacher asks the class to find the probability of choosing a password with one Upper G. Manuela incorrectly says that the probability is 3. Find the correct probability then decide which error Manuela might have made.
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Question content area top right
Part 1
Possible Combinations
​(Upper G​,Upper O​)
​(Upper O​,Upper P​)
​(Upper P​,Upper R​)
​(Upper R​,Upper Y​)
​(Upper Y​,Upper B​)
​(Upper B​,Upper G​)
​(Upper G​,Upper P​)
​(Upper O​,Upper R​)
​(Upper P​,Upper Y​)
​(Upper R​,Upper B​)
​(Upper Y​,Upper G​)
​(Upper B​,Upper O​)
​(Upper G​,Upper R​)
​(Upper O​,Upper Y​)
​(Upper P​,Upper B​)
​(Upper R​,Upper G​)
​(Upper Y​,Upper O​)
​(Upper B​,Upper P​)
​(Upper G​,Upper Y​)
​(Upper O​,Upper B​)
​(Upper P​,Upper G​)
​(Upper R​,Upper O​)
​(Upper Y​,Upper P​)
​(Upper B​,Upper R​)
​(Upper G​,Upper B​)
​(Upper O​,Upper G​)
​(Upper P​,Upper O​)
​(Upper R​,Upper P​)
​(Upper Y​,Upper R​)
​(Upper B​,Upper Y​)
Question content area bottom
Part 1
The probability of choosing a password with one Upper G is
  
enter your response here.
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​(Simplify your​ answer.)

Answer 1

To find the correct probability of choosing a password with one Upper G, we first need to determine the total number of possible 2-character password combinations using the letters G, O, P, R, Y, and B without double letters.

### Step 1: Total Combinations
The total number of combinations can be calculated as follows:
- Since there are 6 letters to choose from and each password consists of 2 letters with no repetitions, we can calculate the total by selecting 2 letters from 6 in different arrangements.

The total number of combinations can be calculated as:
\[
\text{Total combinations} = 6 \text{ choices for the first letter} \times 5 \text{ choices for the second letter} = 30
\]
So there are 30 unique passwords.

### Step 2: Passwords with One Upper G
Now, we need to identify how many of these combinations contain exactly one Upper G. The combinations that include Upper G are:

1. G + O
2. G + P
3. G + R
4. G + Y
5. G + B
6. O + G
7. P + G
8. R + G
9. Y + G
10. B + G

From this, we can see that when G is the first character, we can pair it with O, P, R, Y, or B, giving us 5 combinations. When G is the second character, we can pair it with the same 5 letters (O, P, R, Y, or B), which gives us another 5 combinations.

Thus, we have:
- 5 combinations where G is the first letter.
- 5 combinations where G is the second letter.

Overall, there is a total of \( 5 + 5 = 10 \) passwords that contain exactly one Upper G.

### Step 3: Calculate Probability
To find the probability of choosing a password with one Upper G, we use the formula for probability:
\[
\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{10}{30} = \frac{1}{3}
\]

### Conclusion
Thus, the correct probability of choosing a password with one Upper G is:
\[
\frac{1}{3}
\]

### Error Analysis for Manuela
Manuela likely made an error by mistakenly concluding that the probability was 3. This could happen if she miscounted the number of favorable combinations or misunderstood how to represent probabilities, possibly mixing up the total outcomes with the number of favorable outcomes.