Lets say I have three independent sources and each of them make predictions for the weather tomorrow. The first one says that the probability of rain tomorrow is 0, then the second one says that the probability is 1, and finally the last one says that the probability is 50%. I would like to know the total probability given that information.
If apply the multiplication theorem for independent events I get 0, which doesn't seem correct. Why is not possible to multiply all three if all sources are independent? Is there some Bayesian way to update the prior as I get new information?
Note: This is not homework, is something that I was thinking about.
If you believe the three sources to be equally reliable and that they are the only ones to be considered, then maybe it is reasonable to start with $P(A) = P(B) = P(C) = 1/3.$ Predictions are $P(R|A) = 0,\,$ $P(R|B) = 1,\,$ and $P(R|C) = 1/2.$ Then you might use the law of total probability to get $P(R),\,$ and carry an umbrella tomorrow.