3 letters are typed, without repetition. What is the probability that all 3 will be vowels? Write your answer as a percent. Round your answer to three decimal places.
To find the probability of all 3 letters being vowels, we need to determine the total number of possible outcomes and the number of favorable outcomes.
First, let's determine the total number of possible outcomes. Since there are no repetitions allowed and there are 26 letters in the English alphabet, the first letter can be any of the 5 vowels (A, E, I, O, U), the second letter can be any of the remaining 4 vowels, and the third letter can be any of the remaining 3 vowels.
So, the total number of possible outcomes is 5 * 4 * 3 = 60.
Now, let's determine the number of favorable outcomes. We want all 3 letters to be vowels, so there are no consonants to consider. Therefore, the number of favorable outcomes is simply 3, corresponding to the 3 vowels (A, E, I) remaining to be selected.
The probability of all 3 letters being vowels is then given by the number of favorable outcomes divided by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 3 / 60 = 0.05
To express this as a percentage, we multiply by 100:
0.05 * 100 = 5
Therefore, the probability that all 3 letters will be vowels is 5%.
To find the probability of typing 3 letters without repetition, where all 3 letters are vowels, we need to consider the total number of possible outcomes and the number of favorable outcomes.
Total number of outcomes:
If we type 3 letters without repetition, there are 26 choices for the first letter (since there are 26 English alphabets), and after typing the first letter without repetition, there are 25 choices for the second letter, and 24 choices for the third letter.
Number of favorable outcomes:
Since we want all 3 letters to be vowels, we need to determine the number of vowels in the English alphabet. There are 5 vowels (a, e, i, o, u).
Therefore, there are 5 choices for the first letter (as it must be a vowel), 4 choices for the second letter (as it cannot be the same vowel as the first letter), and 3 choices for the third letter (as it cannot be the same as the first or second letter).
Thus, the number of favorable outcomes is 5 * 4 * 3 = 60.
To find the probability, we divide the number of favorable outcomes by the total number of outcomes:
Probability = Number of favorable outcomes / Total number of outcomes
= 60 / (26 * 25 * 24)
Now we can calculate the probability and round it to three decimal places:
Probability = 60 / (26 * 25 * 24) ≈ 0.0462
Finally, we express the probability as a percentage by multiplying by 100:
Probability as a percent = 0.0462 * 100 ≈ 4.620%
Therefore, the probability that all 3 letters will be vowels is approximately 4.620%.