Final answer:
To find the probability P(−2≤X≤1), one must calculate the differences between the outcomes of rolling two six-sided dice and count the number of results that fall within the range of −2 to 1. Dividing this by the total number of outcomes (36) yields the probability.
Step-by-step explanation:
To determine the probability P(−2≤X≤1), where X is the number on the red cube minus the number on the blue cube, we first consider all the possible outcomes when rolling two six-sided dice. Since each die has 6 faces, there are a total of 6 x 6 = 36 possible outcomes when rolling both dice. We need to count the number of outcomes where the difference between the red and blue dice falls between −2 and 1, inclusive.
To do this, we can create a matrix of the possible differences. For instance, if the red die shows 1 and the blue shows 1 as well, the difference is 0, which falls within our range. We then continue to calculate the differences for each combination of red and blue die rolls and count how many fall within our range of −2 to 1.
After evaluating all combinations, we find that there are several outcomes that satisfy the condition −2≤X≤1. We add these up for the total desired outcomes. Dividing this number by the total number of outcomes (36), gives us the probability P(−2≤X≤1).
Here's a simplified way to visualize it: imagine a 6x6 grid where rows represent red die outcomes and columns represent blue die outcomes. Each cell holds the difference (red - blue). We count all cells where the difference is between -2 and 1.
Finally, we can calculate the probability that the difference is within the specified range. This is the number of favorable outcomes divided by the total number of possible outcomes (36).