Answer: Therefore, the total number of favorable outcomes is 6 * 125 = 750.
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Favorable outcomes / Total outcomes = 750 / 1296 = 125 / 216.
Hence, the probability that no two people sitting next to each other will roll the same number after they each roll the die once is 125/216.
Step-by-step explanation: To calculate the probability that no two people sitting next to each other will roll the same number after they each roll the die once, we can approach the problem using the principle of counting and probability.
First, let's consider the number of possible outcomes for each person rolling a six-sided die independently. Since each person has six possible outcomes (numbers 1 to 6), the total number of possible outcomes for all four people is 6 * 6 * 6 * 6 = 6^4 = 1296.
Now, let's count the number of favorable outcomes where no two people sitting next to each other roll the same number. We can start by fixing the number for the first person arbitrarily, let's say person A rolls a 1. Now, person B sitting next to A has 5 possible numbers to roll (excluding the number rolled by A). Similarly, person C sitting next to B also has 5 possible numbers to roll (excluding the numbers rolled by A and B). Finally, person D sitting next to C also has 5 possible numbers to roll (excluding the numbers rolled by A, B, and C).
Since we arbitrarily fixed person A's roll to be 1, we have 5 * 5 * 5 = 125 favorable outcomes for this particular case. However, this can be done for any number that person A rolls, so we need to multiply this by 6 to consider all possible cases.