Complement Events
In probability theory, complement events are a concept used to describe the relationship between two events
In probability theory, complement events are a concept used to describe the relationship between two events. Given an event A, the complement of A (denoted as A’) is the event that consists of all outcomes or occurrences that are not in A.
To understand complement events, let’s consider an example. Suppose we roll a fair six-sided die. Event A could be defined as rolling an even number, which includes the outcomes {2, 4, 6}. The complement of event A (A’) would then be rolling an odd number, which includes the outcomes {1, 3, 5}.
Here are some important properties of complement events:
1. An outcome or occurrence either belongs to event A or its complement A’, but not both. In our example, rolling a 2 belongs to event A (even number), and rolling a 1 belongs to its complement A’ (odd number).
2. The union of event A and its complement A’ is the entire sample space. The sample space in our example would be {1, 2, 3, 4, 5, 6}, and event A (even number) and its complement A’ (odd number) together cover all possible outcomes.
3. The intersection of event A and its complement A’ is the empty set. In other words, there are no outcomes that simultaneously belong to event A and its complement A’. In our example, there are no numbers that are both even and odd.
Complement events are particularly useful in probability calculations. If we know the probability of event A, we can easily obtain the probability of its complement A’ by subtracting the probability of A from 1. This is based on the fact that the sum of the probabilities of two complementary events is always 1.
For example, if the probability of rolling an even number (event A) is 1/2, then the probability of rolling an odd number (complement A’) is 1 – 1/2 = 1/2 as well.
Overall, complement events allow us to describe the occurrence/non-occurrence relationship between two events, providing a useful tool for probability analysis.
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