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$)=\frac{1}{5}$ $p(b$ or d on the second roll, given that the first roll was a consonant$)$ > enter the answer in the space provided. use numbers instead of words.

Explanation:

Step1: Identify consonants on first - roll

Consonants among A, B, C, D, E are B, C, D. So there are 3 consonants. The total number of outcomes for the first roll is 5. The number of total outcomes for two - roll combinations is \(n = 5\times5=25\). The number of outcomes where the first roll is a consonant is \(m_1 = 3\times5 = 15\) (because for each of the 3 consonants on the first roll, there are 5 possible outcomes on the second roll).

Step2: Identify favorable outcomes

The favorable outcomes (first roll is a consonant and second roll is B or D) are: When the first roll is B, the favorable second - roll outcomes are B and D; when the first roll is C, the favorable second - roll outcomes are B and D; when the first roll is D, the favorable second - roll outcomes are B and D. The number of favorable outcomes \(m_2=3\times2 = 6\).

Step3: Calculate the conditional probability

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). In terms of counting outcomes, \(P(\text{B or D on second roll}|\text{first roll was a consonant})=\frac{\text{Number of outcomes where first roll is a consonant and second roll is B or D}}{\text{Number of outcomes where first roll is a consonant}}\). So \(P=\frac{6}{15}=\frac{2}{5}\).

Answer:

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