Using the axioms of probability, prove P(𝐴𝑐) = 1 − 𝑃(𝐴)

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To prove the equation P(𝐴𝑐) = 1 − 𝑃(𝐴) using the axioms of probability, we can follow these steps:

Step 1: Start with the definition of the complement of an event 𝐴. The complement of 𝐴, denoted as 𝐴𝑐, is the set of all outcomes that are not in 𝐴.

Step 2: According to the axioms of probability, the probability of the sample space, denoted as 𝑆, is equal to 1. In other words, P(𝑆) = 1.

Step 3: Since 𝐴 and 𝐴𝑐 together make up the entire sample space, we can say that 𝐴 ∪ 𝐴𝑐 = 𝑆.

Step 4: Using the axiom of probability that states the probability of the union of two mutually exclusive events is equal to the sum of their individual probabilities, we can write P(𝐴 ∪ 𝐴𝑐) = P(𝐴) + P(𝐴𝑐).

Step 5: Since 𝐴 and 𝐴𝑐 are mutually exclusive (i.e., they cannot occur at the same time), the probability of their union is equal to the probability of the sample space, which is 1. Therefore, P(𝐴 ∪ 𝐴𝑐) = P(𝑆) = 1.

Step 6: Substituting the values from steps 3 and 5 into step 4, we get 1 = P(𝐴) + P(𝐴𝑐).

Step 7: Rearranging the equation from step 6, we can write P(𝐴𝑐) = 1 − P(𝐴).

Therefore, using the axioms of probability, we have proven that P(𝐴𝑐) = 1 − P(𝐴).