We are tasked with finding the probability that at least one married couple sits next to each other when three married couples (6 guests altogether) sit randomly at a round table.

Step 1: Total Number of Arrangements

Since the table is round, we can fix one person’s seat to remove rotational symmetry. This leaves 5 remaining people to be seated randomly. Therefore, the total number of possible seating arrangements is: $5! = 120.$

Step 2: Number of Favorable Arrangements (at least one couple is seated together)

To solve this, we use complementary counting, i.e., we first count the number of ways in which no couple is seated together, and then subtract that from the total number of arrangements.

Step 2.1: Treat couples as seated separately

If no couple is seated together, we calculate the number of derangements of the couples, meaning we count the number of ways in which no couple sits next to each other.

To use the derangement formula for this problem, we first treat each couple as a unit and subtract the number of ways to seat them incorrectly. However, this calculation is complex, and we simplify it by using a general formula or enumeration strategy for small cases like this.

For simplicity, though, we can calculate the specific arrangements that allow each spouse to be seated at a different seat, resulting in