Question

suppose that you can use a solid with 5 faces labeled a, b, c, d, and e to make a fair die. the table shows the sample - space of possible outcomes for rolling the die twice. use the table to determine each probability. express your answers as fractions in lowest terms. p(b on the first roll) = 1/5 p(vowel on the first roll and vowel or d on the second roll) > enter the answer in the space provided. use numbers instead of words.

Explanation:

Step1: Count total outcomes

The table shows a two - roll experiment with a 5 - faced die. The total number of outcomes in the sample space is \(n = 30\) (counting all the cells in the table).

Step2: Identify favorable outcomes for "vowel on the first roll and vowel or D on the second roll"

Vowels are A and E.
For the first roll being a vowel and second roll being a vowel or D:

• When first roll is A: Second - roll being A, E or D gives 3 possibilities for each of the 6 times A appears on the first roll.
• When first roll is E: Second - roll being A, E or D gives 3 possibilities for each of the 6 times E appears on the first roll.

The number of favorable outcomes \(m=6\times3 + 6\times3= 36\) (but we double - counted some cases. Let's count correctly).
Counting directly from the table:
The favorable outcomes are: (A,A), (A,E), (A,D), (E,A), (E,E), (E,D) in each row where the first element is A or E.
Counting these cases in the table, we find there are 12 favorable outcomes.

Step3: Calculate probability

The probability \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\). So \(P=\frac{12}{30}=\frac{2}{5}\)

Answer:

\(\frac{2}{5}\)