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) =? p(consonant on the second roll, given that the first roll was a vowel) > enter the answer in the space provided. use numbers instead of words.

Explanation:

Step1: Determine total number of outcomes

The die is rolled twice, and since it has 5 faces, the total number of outcomes in the sample - space is \(n(S)=5\times5 = 25\).

Step2: Identify outcomes where first - roll is B

The outcomes where the first roll is B are (B,A), (B,B), (B,C), (B,D), (B,E), so \(n(\text{first roll is B}) = 5\).

Step3: Calculate probability

The probability \(P(\text{B on the first roll})\) is given by the formula \(P=\frac{n(\text{event})}{n(S)}\). So \(P(\text{B on the first roll})=\frac{5}{25}=\frac{1}{5}\).

Step4: Identify vowels and consonants

Vowels are A and E, consonants are B, C, D.

Step5: Identify outcomes where first roll is a vowel and second roll is a consonant

The outcomes where the first roll is a vowel (A or E) and the second roll is a consonant (B, C, D) are: (A,B), (A,C), (A,D), (E,B), (E,C), (E,D). So \(n = 6\).

Step6: Identify outcomes where first roll is a vowel

The outcomes where the first roll is a vowel (A or E) are: (A,A), (A,B), (A,C), (A,D), (A,E), (E,A), (E,B), (E,C), (E,D), (E,E), so \(n(\text{first roll is a vowel}) = 10\).

Step7: Calculate conditional probability

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). In terms of counts, \(P(\text{consonant on the second roll}|\text{first roll was a vowel})=\frac{n(\text{first roll is vowel and second roll is consonant})}{n(\text{first roll is vowel})}=\frac{6}{10}=\frac{3}{5}\).

Answer:

\(P(\text{B on the first roll})=\frac{1}{5}\), \(P(\text{consonant on the second roll}|\text{first roll was a vowel})=\frac{3}{5}\)