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To update the probability of rain given the observation of cloudiness, you apply Bayes' theorem. You substitute the supplied prior probability of rain, the likelihood of cloudiness given rain, and the marginal probability of cloudiness into the formula.

Conditions: Cloudiness is the observed event (B).; Rain is the event of interest (A).; Prior probabilities and likelihoods are supplied.; $P(\text{Cloudiness}) > 0$.

To update the probability of rain given the observation of cloudiness, you apply Bayes' theorem. You substitute the supplied prior probability of rain, the likelihood of cloudiness given rain, and the marginal probability of cloudiness into the formula.

Conditions: Cloudiness is the observed event (B).; Rain is the event of interest (A).; Prior probabilities and likelihoods are supplied.; $P(\text{Cloudiness}) > 0$.

The derivation of Bayes' theorem requires that the conditioning probabilities be nonzero. Specifically, when deriving $P(A|B)$ from the intersection, we divide by $P(B)$, so $P(B)$ must be positive.

Conditions: Deriving Bayes' formula from conditional probability definitions.; $P(B) > 0$ for $P(A|B)$ to be defined.; $P(A) > 0$ for $P(B|A)$ to be defined.

The derivation of Bayes' theorem requires that the conditioning probabilities be nonzero. Specifically, when deriving $P(A|B)$ from the intersection, we divide by $P(B)$, so $P(B)$ must be positive.

Conditions: Deriving Bayes' formula from conditional probability definitions.; $P(B) > 0$ for $P(A|B)$ to be defined.; $P(A) > 0$ for $P(B|A)$ to be defined.