A pair of dice is rolled. Is the event of rolling a sum of 2, 4, 6, or 8 independent from rolling a sum of 3, 5, 7, or 9?
Yes, unless the dice are "fixed," each roll is independent of other rolls.
To determine if the event of rolling a sum of 2, 4, 6, or 8 is independent from rolling a sum of 3, 5, 7, or 9, we need to consider the probabilities of each event separately and then compare them.
Let's start by calculating the probability of rolling a sum of 2, 4, 6, or 8. To do this, we need to find all the possible combinations of two dice that result in those sums.
For a sum of 2, there is only one possible combination: (1, 1).
For a sum of 4, there are three possible combinations: (1, 3), (2, 2), and (3, 1).
For a sum of 6, there five possible combinations: (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1).
For a sum of 8, there are also five possible combinations: (2, 6), (3, 5), (4, 4), (5, 3), and (6, 2).
In total, there are 1 + 3 + 5 + 5 = 14 possible combinations that result in a sum of 2, 4, 6, or 8. Since each die has 6 sides and can result in 6 possible outcomes, the total number of outcomes for rolling two dice is 6 * 6 = 36.
Therefore, the probability of rolling a sum of 2, 4, 6, or 8 is 14/36, which simplifies to 7/18.
Next, let's calculate the probability of rolling a sum of 3, 5, 7, or 9. We can use a similar approach as before.
For a sum of 3, there are two possible combinations: (1, 2) and (2, 1).
For a sum of 5, there are four possible combinations: (1, 4), (2, 3), (3, 2), and (4, 1).
For a sum of 7, there are six possible combinations: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1).
For a sum of 9, there are also four possible combinations: (3, 6), (4, 5), (5, 4), and (6, 3).
In total, there are 2 + 4 + 6 + 4 = 16 possible combinations that result in a sum of 3, 5, 7, or 9.
Therefore, the probability of rolling a sum of 3, 5, 7, or 9 is 16/36, which simplifies to 4/9.
Now, to determine if these events are independent, we need to compare the probabilities. If the probability of one event does not change based on the occurrence of the other event, then they are independent.
In our case, the probability of rolling a sum of 2, 4, 6, or 8 is 7/18, and the probability of rolling a sum of 3, 5, 7, or 9 is 4/9.
Since the probabilities are not equal, we can conclude that the event of rolling a sum of 2, 4, 6, or 8 is not independent from rolling a sum of 3, 5, 7, or 9.