Question

a single die is rolled twice. find the probability of rolling an odd number and a number greater than 4 in either order. the probability of rolling an odd number and a number greater than 4 is (type an integer or a simplified fraction.)

Explanation:

Step1: Determine sample space size

When a die is rolled twice, the sample - space has \(n(S)=6\times6 = 36\) possible outcomes.

Step2: Identify favorable outcomes

The odd numbers on a die are \(1,3,5\) and the numbers greater than \(4\) are \(5,6\).
Case 1: First roll is odd and second roll is greater than 4.
If the first roll is odd (\(3\) possibilities: \(1,3,5\)) and the second roll is greater than 4 (\(2\) possibilities: \(5,6\)), the number of outcomes is \(3\times2 = 6\).
Case 2: First roll is greater than 4 and second roll is odd.
If the first roll is greater than 4 (\(2\) possibilities: \(5,6\)) and the second roll is odd (\(3\) possibilities: \(1,3,5\)), the number of outcomes is \(2\times3=6\). But we have double - counted the outcome \((5,5)\) once.
The total number of favorable outcomes \(n(A)=6 + 6-1= \frac{1}{3}\).
The probability \(P(A)=\frac{n(A)}{n(S)}=\frac{5}{18}\).

Answer:

\(\frac{5}{18}\)