Question

you roll a six - sided die twice. what is the probability of rolling an even number and then an odd number?

Explanation:

Step1: Calculate the probability of rolling an even number

A six - sided die has 6 possible outcomes \(\{1,2,3,4,5,6\}\). The even numbers are \(2\), \(4\), \(6\). So the number of favorable outcomes for rolling an even number is \(n_1 = 3\). The probability of rolling an even number \(P(\text{even})=\frac{3}{6}=\frac{1}{2}\).

Step2: Calculate the probability of rolling an odd number

The odd numbers on a six - sided die are \(1\), \(3\), \(5\). So the number of favorable outcomes for rolling an odd number is \(n_2 = 3\). The probability of rolling an odd number \(P(\text{odd})=\frac{3}{6}=\frac{1}{2}\).

Step3: Use the multiplication rule for independent events

Since the two rolls of the die are independent events, the probability of rolling an even number and then an odd number is \(P = P(\text{even})\times P(\text{odd})\). Substitute \(P(\text{even})=\frac{1}{2}\) and \(P(\text{odd})=\frac{1}{2}\) into the formula: \(P=\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}\).

Answer:

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