To find the probability of choosing a password with no Ys, let's first identify the total sample space and the number of passwords without Ys.
Step 1: Determine the total number of possible combinations
The letters are: Y, O, R, B, P, G.
Since double letters are not allowed, we have 6 letters total. For a 2-character password, we can choose the first character in 6 ways and the second in 5 ways, giving us:
\[ \text{Total combinations} = 6 \times 5 = 30 \]
Step 2: Determine the number of combinations with no Ys
If we exclude the letter Y, the available letters are: O, R, B, P, G. This gives us 5 letters to choose from.
We can choose the first character in 5 ways and the second character in 4 ways (since we cannot repeat characters):
\[ \text{Combinations with no Ys} = 5 \times 4 = 20 \]
Step 3: Calculate the probability of choosing a password with no Ys
The probability is given by the formula:
\[ P(\text{no Ys}) = \frac{\text{Number of combinations with no Ys}}{\text{Total combinations}} = \frac{20}{30} \]
This simplifies to:
\[ P(\text{no Ys}) = \frac{2}{3} \]
Conclusion
The correct probability of choosing a password with no Ys is \( \frac{2}{3} \).
error analysis
Manuela incorrectly stated that the probability is three halves, which suggests that she may have:
- Miscounted the total number of combinations or combinations without Ys.
- Confused the structure of fractions leading to an incorrect result of greater than 1.
In reality, the correct probability indicates that there are more combinations that contain Y than there are without it, hence the probability cannot exceed 1.