91Ó°ÊÓ

Step 1: Calculate the sample space. There are two dice with six faces each. The total number of possible outcomes when rolling two dice is \(6 \times 6 = 36\). Step 2: Calculate the complementary probabilities The only way that neither die shows a 6 is if both dice show a number from 1 to 5. The probability of not getting a 6 on one die is \(\frac{5}{6}\). Step 3: Multiply the complementary probabilities Since the events are independent, multiply the probabilities of not getting a 6 on each die: \(\frac{5}{6} \times \frac{5}{6} = \frac{25}{36}\). Step 4: Find the probability of the desired event The probability of at least one 6 is equal to 1 minus the probability of getting no 6s: \( 1 - \frac{25}{36} = \frac{11}{36}\). So, the probability of getting at least one 6 when rolling two dice is \(\frac{11}{36}\).

Step 1: Calculate the new sample space Given that the two dice show different numbers, there are 6 options for the first die and 5 options for the second die (since it cannot show the same number as the first die). This results in a sample space of \(6 \times 5 = 30\). Step 2: Calculate the complementary probabilities with constraints In the new sample space, neither die can obtain a 6. The probability of not getting a 6 on the first die remains the same, \(\frac{5}{6}\), but for the second die, it only has five available outcomes now, so the probability of not getting a 6 on it is \(\frac{4}{5}\). Step 3: Multiply the complementary probabilities with constraints Again, since the events are independent, multiply the probabilities of not getting a 6 on each die: \(\frac{5}{6} \times \frac{4}{5} = \frac{20}{30}\). Step 4: Find the probability of the desired event with constraints Using a complementary approach, the probability of getting at least one 6 with the constraint is equal to 1 minus the probability of no 6s: \(1 - \frac{20}{30} = \frac{10}{30}\). So, the probability of getting at least one 6 when rolling two dice with the constraint that the two faces are different is \(\frac{10}{30}\) or \(\frac{1}{3}\).