To solve for the probability of choosing two vowels from a bag containing 12 Hawaiian letters (5 of which are vowels) where each letter is replaced before the next pick, we can follow these steps:
1. Determine the probability of choosing a vowel on the first pick:
- There are 5 vowels and a total of 12 letters.
- Therefore, the probability of picking a vowel on the first draw is:
[tex]\[
\frac{5}{12}
\][/tex]
2. Determine the probability of choosing a vowel on the second pick:
- Since the letter is replaced, the scenario remains the same for the second draw.
- Thus, the probability of picking a vowel again is also:
[tex]\[
\frac{5}{12}
\][/tex]
3. Calculate the combined probability of both events happening (choosing a vowel on the first pick AND a vowel on the second pick):
- Since these are independent events (because of replacement), we multiply the probabilities of the individual events:
[tex]\[
\left( \frac{5}{12} \right) \times \left( \frac{5}{12} \right)
\][/tex]
[tex]\[
\frac{5}{12} \times \frac{5}{12} = \frac{25}{144}
\][/tex]
Therefore, the probability that two vowels are chosen is:
[tex]\[
\boxed{\frac{25}{144}}
\][/tex]
By comparing this with the given options, we see that our answer matches option (b) [tex]$\frac{25}{144}$[/tex].
