Final answer:
The inverse image of the complement in the context of probability and sets refers to finding the original event given its complement. The probabilities of an event and its complement sum up to 1. Therefore, if the probability of an event's complement is known, the probability of the event itself is 1 minus the probability of the complement.
Step-by-step explanation:
The concept of the inverse image of the complement relates to set theory and probability in mathematics. To discuss the inverse image of the complement, one must understand the concepts of events, their complements, and probability spaces.
To start, let's consider a simple event №(A), which represents a certain outcome within a sample space. The complement of event A, denoted by №(A'), consists of all outcomes that are not in A. If we have a function (f) that assigns probabilities to these events, such as f(A) representing the probability of event A happening, then f(A') would represent the probability of the complement of A happening, i.e., A not happening.
When we refer to the inverse image under a function, we are usually talking about elements in the domain being mapped back to the codomain. However, in the context of probability, the "inverse image" is not used in the traditional sense. Here, P(A) + P(A') = 1, since the probabilities of an event and its complement must sum up to 1. Thus, to find the inverse image of the complement, you would essentially find the event whose probability, when added to the probability of the complement, equals 1.
Considering the provided information, let's use the example where B is the event of a child having brown hair. The inverse image of the complement of B would refer to finding the original event B given its complement. If the probability of the complement of B (i.e., not having brown hair) is given, then the probability of B (having brown hair) can be found by subtracting the probability of the complement from 1.