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. sample problem $p(b$ on the first roll$)=\frac{1}{5}$ $p($vowel on the second roll, given that the first roll was not an a) > enter the answer in the space provided. use numbers instead of words.

Explanation:

Step1: Determine total number of non - A first - roll outcomes

The first roll has 5 possible outcomes. Since we are considering the cases where the first roll was not an A, there are 4 non - A outcomes. For each of these 4 non - A first - roll outcomes, there are 5 possible second - roll outcomes. So the total number of outcomes where the first roll was not an A is \(4\times5 = 20\).

Step2: Determine number of non - A first - roll and vowel second - roll outcomes

Vowels among the labels are A and E. Since the first roll is not A, we consider the cases where the first roll is B, C, D, or E. For each of these 4 first - roll non - A outcomes, there are 2 vowel (A and E) second - roll outcomes. So the number of non - A first - roll and vowel second - roll outcomes is \(4\times2=8\).

Step3: Calculate the conditional probability

The formula for conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). In the case of equally - likely outcomes, if A is the event of getting a vowel on the second roll and B is the event of not getting an A on the first roll, then \(P(\text{vowel on second roll}|\text{first roll was not A})=\frac{\text{Number of non - A first - roll and vowel second - roll outcomes}}{\text{Total number of non - A first - roll outcomes}}\). So the probability is \(\frac{8}{20}=\frac{2}{5}\).

Answer:

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