Question

a single six - sided die is rolled twice. what is the probability of rolling an odd number on the first roll and an even number on the second roll?
a. $\frac{1}{12}$
b. $\frac{1}{24}$
c. $\frac{1}{4}$
d. $\frac{1}{2}$

Explanation:

Step1: Calculate probability of odd on first roll

A six - sided die has 6 possible outcomes \(\{1,2,3,4,5,6\}\). The odd numbers are \(1,3,5\), so the number of favorable outcomes for rolling an odd number is \(n_{1}=3\). The probability \(P(\text{odd})=\frac{n_{1}}{N}=\frac{3}{6}=\frac{1}{2}\).

Step2: Calculate probability of even on second roll

The even numbers on a six - sided die are \(2,4,6\), so the number of favorable outcomes for rolling an even number is \(n_{2} = 3\). The probability \(P(\text{even})=\frac{n_{2}}{N}=\frac{3}{6}=\frac{1}{2}\).

Step3: Use the multiplication rule for independent events

Since the two rolls of the die are independent events (the outcome of the first roll does not affect the outcome of the second roll), the probability of both events occurring is \(P = P(\text{odd})\times P(\text{even})\). Substitute \(P(\text{odd})=\frac{1}{2}\) and \(P(\text{even})=\frac{1}{2}\) into the formula: \(P=\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}\).

Answer:

C. \(\frac{1}{4}\)