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:
- G + O
- G + P
- G + R
- G + Y
- G + B
- O + G
- P + G
- R + G
- Y + G
- 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.