Question

the hawaiian language has 12 letters, five vowels and seven consonants. each of the 12 hawaiian letters are written on a slip of paper and placed in the bag. a letter is randomly chosen from the bag and then replaced. then, a second letter is randomly chosen from the bag. what is the probability that two vowels are chosen?
\\(\frac{5}{72}\\)
\\(\frac{25}{144}\\)
\\(\frac{7}{12}\\)
\\(\frac{5}{6}\\)

Explanation:

Step1: Calculate probability of first - vowel

The probability of choosing a vowel on the first draw is the number of vowels divided by the total number of letters. There are 5 vowels and 12 total letters, so the probability $P_1=\frac{5}{12}$.

Step2: Calculate probability of second - vowel

Since the letter is replaced, the probability of choosing a vowel on the second draw is also $\frac{5}{12}$ because the situation is the same as the first draw.

Step3: Calculate probability of both events

For independent events, the probability of both events occurring is the product of their individual probabilities. So $P = P_1\times P_2=\frac{5}{12}\times\frac{5}{12}=\frac{25}{144}$.

Answer:

$\frac{25}{144}$