a single die is rolled twice. find the probability of rolling an even number the first time and a number…

Explanation:

Step1: Calculate the probability of rolling an even number

A die has 6 faces: (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 formula is (P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}). So (P_1=\frac{3}{6}=\frac{1}{2}).

Step2: Calculate the probability of rolling a number greater than 5

The numbers greater than 5 on a die is 6. So the number of favorable outcomes for rolling a number greater than 5 is (n_2 = 1). Using the probability formula (P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}), we get (P_2=\frac{1}{6}).

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), and for independent events (A) and (B), (P(A\cap B)=P(A)\times P(B)). Let (A) be the event of rolling an even number on the first roll and (B) be the event of rolling a number greater than 5 on the second roll. Then (P = P_1\times P_2). Substitute (P_1=\frac{1}{2}) and (P_2=\frac{1}{6}) into the formula: (P=\frac{1}{2}\times\frac{1}{6}).

Answer:

(\frac{1}{12})