What does the rule of complements state concerning probabilities?

What does the rule of complements state concerning probabilities?

• A. The probability of an event is twice the probability of its complement
• B. The probability of an event occurring equals the probability of it not occurring
• C. The probability of an event occurring equals one minus the probability of it not occurring
• D. The probability of two unrelated events happening together is zero

Answer: (C)

The rule of complements is fundamental in probability theory, and it states that the probability of an event occurring is equal to one minus the probability of the event not occurring. This can be mathematically expressed as \( P(A) = 1 - P(A') \), where \( P(A) \) is the probability of event A occurring and \( P(A') \) is the probability of A not occurring (its complement). This relationship is particularly useful when calculating probabilities when the direct computation of the event of interest is complicated, allowing us to instead focus on its complement. For instance, if it is easier to determine the likelihood of a certain scenario not happening, using this rule enables an efficient way to find the probability of the original event. The other options do not accurately represent the rule of complements. For example, stating that the probability of an event is twice the probability of its complement misrepresents the relationship entirely, while claiming that the probabilities of an event occurring and not occurring are equal ignores the fundamental concept that these two outcomes are mutually exclusive and collectively exhaustive, summing to one. The statement about unrelated events also does not pertain to the rule of complements, as it addresses a different aspect of probability theory entirely.

The rule of complements is fundamental in probability theory, and it states that the probability of an event occurring is equal to one minus the probability of the event not occurring. This can be mathematically expressed as ( P(A) = 1 - P(A') ), where ( P(A) ) is the probability of event A occurring and ( P(A') ) is the probability of A not occurring (its complement).

This relationship is particularly useful when calculating probabilities when the direct computation of the event of interest is complicated, allowing us to instead focus on its complement. For instance, if it is easier to determine the likelihood of a certain scenario not happening, using this rule enables an efficient way to find the probability of the original event.

The other options do not accurately represent the rule of complements. For example, stating that the probability of an event is twice the probability of its complement misrepresents the relationship entirely, while claiming that the probabilities of an event occurring and not occurring are equal ignores the fundamental concept that these two outcomes are mutually exclusive and collectively exhaustive, summing to one. The statement about unrelated events also does not pertain to the rule of complements, as it addresses a different aspect of probability theory entirely.