According to the total probability rule, P(A) equals the sum of P(A $\cap$ B) and P(A $\cap$ B^c), and is considered conditional on two mutually exclusive and exhaustive events independent of an experiment.

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The total probability rule states that if $B_1, B_2, \ldots, B_n$ are mutually exclusive and exhaustive events, then $P(A) = P(A \mid B_1)P(B_1) + P(A \mid B_2)P(B_2) + \ldots + P(A \mid B_n)P(B_n)$. It does not simply add $P(A \cap B)$ and $P(A \cap B^c)$; it involves conditional probabilities multiplied by the probabilities of the conditions.

According to the Total Probability Rule, P(A) Equals the Sum ∩\cap∩

According to the Total Probability Rule, P(A) Equals the Sum $\cap$