Six individuals are sitting in a circle. I have six cards, three red and three blue. If I randomly hand each one a card, what is the probability that the colors would perfectly alternate, that is no two adjacent people have the same color

The probability is 1/10.

Step-by-step explanation:

Without loss of generality, the first card is blue. Then choose a red one with probability 3/5, a blue one with probability 2/4=1/2, a red one with probability 2/3, and the last blue one with probability 1/2, followed by the last red one.

The result is 2/5⋅(1/2)^2=1/10.

OR

There are actually 6!/(3!3!)=720/36=20 ways, so the probability is 2/20=1/10.