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Bayes' formula is derived by expressing the joint probability of the intersection of events A and B, $P(A \cap B)$, in two different ways using the definition of conditional probability. By equating these two expressions, $P(A|B)P(B)$ and $P(B|A)P(A)$, and then dividing by $P(B)$, we isolate $P(A|B)$ to obtain Bayes' theorem.

Conditions: $P(B) > 0$ to allow division.; Events A and B are defined in the same probability space.

Bayes' formula is derived by expressing the joint probability of the intersection of events A and B, $P(A \cap B)$, in two different ways using the definition of conditional probability. By equating these two expressions, $P(A|B)P(B)$ and $P(B|A)P(A)$, and then dividing by $P(B)$, we isolate $P(A|B)$ to obtain Bayes' theorem.

Conditions: $P(B) > 0$ to allow division.; Events A and B are defined in the same probability space.

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.