Let’s assume prediction markets are efficient and you didn’t already possess any relevant information that you weren’t trading on beforehand. Then you should treat the market odds as a prior and your die roll as evidence, in exactly the way you always do Bayesian updates. In this case, that means it looks like that gives you a posterior probability of 5⁄14 each for the die being weighted in favor of 3 or 6, and 1⁄14 for each of the other possibilities. Contrary to what other commenters were saying, it doesn’t matter what information led to the market odds under these assumptions.
Let’s assume prediction markets are efficient and you didn’t already possess any relevant information that you weren’t trading on beforehand. Then you should treat the market odds as a prior and your die roll as evidence, in exactly the way you always do Bayesian updates. In this case, that means it looks like that gives you a posterior probability of 5⁄14 each for the die being weighted in favor of 3 or 6, and 1⁄14 for each of the other possibilities. Contrary to what other commenters were saying, it doesn’t matter what information led to the market odds under these assumptions.