Question

1. you are trying to guess a three-letter password that uses only the letters a, e, i, o, u, and y. letters can be used more than once. find the probability that you pick the correct password “you.”

1. you are trying to guess a three-letter password that uses only the letters a, e, i, o, u, and y. letters cannot be used more than once. find the probability that you pick the correct password “aie.”

Explanation:

Problem 3

Step1: Determine total possible passwords

There are 6 letters (A, E, I, O, U, Y) and we're creating a 3 - letter password with repetition allowed. For each of the 3 positions, we have 6 choices. So by the multiplication principle, the total number of possible passwords is \(6\times6\times6 = 6^{3}=216\).

Step2: Determine favorable outcomes

There is only 1 favorable outcome, which is the password "YOU".

Step3: Calculate probability

The probability \(P\) of an event is given by the formula \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\). So \(P = \frac{1}{216}\).

Step1: Determine total possible passwords

There are 6 letters (A, E, I, O, U, Y) and we're creating a 3 - letter password with no repetition. For the first position, we have 6 choices, for the second position we have 5 remaining choices (since we can't repeat the first letter), and for the third position we have 4 remaining choices. By the multiplication principle, the total number of possible passwords is \(6\times5\times4=\frac{6!}{(6 - 3)!}=120\).

Step2: Determine favorable outcomes

There is only 1 favorable outcome, which is the password "AIE".

Step3: Calculate probability

Using the probability formula \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\), we get \(P=\frac{1}{120}\).

Answer:

\(\frac{1}{216}\)

Problem 4