# What is a complement?

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Probability for Statistics and Data ScienceProbability for improved business decisions: Introduction, Combinatorics, Bayesian Inference, Distributions

03:40:27 of on-demand video • Updated January 2020

- Understand probability theory
- Discover Combinatorics
- Learn how to use and interpret Bayesian Notation
- Different types of distributions variables can follow

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Welcome back folks. Before we continue to the next section of this course let's talk about some of the characteristics of probabilities and events. For starters let's define what a compliment is. Simply put a complement of an event is everything the event is not as the name suggests the complement helps complete the rest of the sample space to calculate the probability of the complement of an event. We need to set up a few things. For starters if we add the probabilities of different events we get their sum of probabilities. Now if we add up all the possible outcomes of an event we should always get one. Remember that having a probability of one is the same as being 100 percent certain we are going to explain why this is true with several examples. OK imagine you are tossing a coin when it falls. We are guaranteed to get either heads or tails. Therefore if we account for the sum of all the probabilities of getting heads or tails we have completely exhausted all possible outcomes. We have accounted for the entire sample space so we are 100 percent certain to get one of the two since we are certain one of these will occur. The sum of their probabilities should be 1. So what would it mean if we have a sum of probabilities greater than one recall the probability of one expresses absolute certainty by definition we cannot be any surer than being absolutely sure. So a probability of one point five does not make intuitive sense instances where we can get such a sum of probabilities is when some of the assumed outcomes can occur simultaneously. This means we are double counting some of the actual possible outcomes we will learn how to deal with such issues when we introduce Bayesian notation less than an hour from now. Now another peculiar case is if we end up with a sum of probabilities less than one then we have surely not accounted for one or several possible outcomes. Probability expresses the likelihood of an event occurring so any probability less than 1 is not guaranteed to occur. Therefore there must be some part of the sample space we have not yet accounted for. Great before we move on we want to tell you that all events have complements and we denote them by adding an apostrophe for example the complement of the event a is denoted as a apostrophe. It is also worth noting that the complement of a complement is the event itself so a apostrophe apostrophe would equal a. Now imagine if you were rolling a standard six sided die and want to roll and even number the opposite of that would be not rolling and even number which is the same as wanting to roll an odd number complements are often used when the event we want to occur is satisfied by many outcomes. For example you want to know the probability of rolling a one to four five or six. That is the same as the probability of not rolling a three. This concept is extremely useful and will definitely come in handy during the next section. We already said that the sum of the probabilities of all possible outcomes equals one. So you can probably guess how we calculate complements the probability of the inverse equals 1 minus the probability of the event itself to make sure you understand the notion well we will look at the example we mentioned earlier the sum of probabilities of getting one two 4 five or six is equal to the sum of the separate probabilities the likelihood of each outcome is equal to one sixth so the sum of their probabilities adds up to five sixth. Now another way of describing getting one two four five or six is not getting a three. Let us calculate the probability of not getting a three. This is the complement of getting a three so we know that the two should add up to 1. Therefore the probability of not getting a three equals one minus the probability of getting a three. We know that p of three equals one sixth so the probability of not getting three is equal to one minus one sixth. Therefore the probability of not getting three is five sixths. This shows that the probability of getting one two four five or six is equal to the probability of not getting a three Now that we have explained the basic probability notions. Let us get back to the lottery example we explored in the first lesson. In fact this will happen throughout a series of lessons where we'll introduce you to the field of combinatorics. We are going to talk about variations permutations and combinations explaining what each of those terms means when to use them and how to compute them. Keep up the good work. See you in the next video. And thanks for watching.