Final answer:
When two dice are rolled, there are 36 possible outcomes. The probability of getting a sum of 6 is 5/36, the probability of getting the same numbers is 1/6, and the probability of both getting a sum of 6 and the same numbers is 1/36.
Step-by-step explanation:
To list all the outcomes of rolling two dice in a two-way table, you need to consider all the possible combinations of numbers that can appear on each die. For example, the first die can show a 1, 2, 3, 4, 5, or 6, and the second die can also show a 1, 2, 3, 4, 5, or 6. So, there are 6 possible outcomes for the first die and 6 possible outcomes for the second die, resulting in a total of 6 x 6 = 36 outcomes.
To find the probability that the sum of two numbers on the dice will be 6, you need to determine the number of favorable outcomes (the sum of two numbers is 6) and divide it by the total number of possible outcomes (36). In this case, the favorable outcomes are (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1), which gives a total of 5 favorable outcomes. Therefore, the probability is 5/36.
To find the probability that the numbers on the dice are the same, you need to determine the number of favorable outcomes (both dice showing the same number) and divide it by the total number of possible outcomes (36). In this case, the favorable outcomes are (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), and (6, 6), which gives a total of 6 favorable outcomes. Therefore, the probability is 6/36, which simplifies to 1/6.
To find the probability that the sum of two numbers on the dice will be 6 and the numbers on the dice are the same, you need to determine the number of favorable outcomes (both dice showing the same number that sums up to 6) and divide it by the total number of possible outcomes (36). In this case, the only favorable outcome is (3, 3). Therefore, the probability is 1/36.