Complements: Videos & Practice Problems

To find the probability of an event not occurring, known as the complement, use the formula: P(¬A) = 1 - P(A). For example, when rolling a six-sided die, the probability of not rolling a four is calculated as P(¬A) = 1 - (1/6) = 5/6. Similarly, in a standard deck of cards, the probability of not drawing a queen is P(¬A) = 1 - (4/52) = 48/52 or 0.92. This principle emphasizes that the sum of the probabilities of an event and its complement equals one, reinforcing the Complement Rule in probability.

Complementary Events

Hey, everyone. We just learned how to find the probability that some event will happen, like, say, rolling a six-sided die and getting a four. But what about the probability that some event will not happen? Well, this is something that you'll actually be asked explicitly to calculate, and it may sound like it's going to be tricky, but you can actually do it using something that we already know. So here, I'm going to walk you through how to calculate the probability that some event will not happen by simply using the probability that it will.

So let's go ahead and get started here. Now, all of the possible outcomes where some event does not happen actually have their own special name, and it's referred to as the complement of that event. So when looking at my dice roll here, if I consider all of the possible outcomes of not rolling a four, like rolling a one, two, three, five, or six, all of these outcomes together represent the complement of rolling a four. If we refer to our event as A, we can use a special notation to denote the complement of A. You may see this written as A′ or with a line over it or with a little symbol in front of it that just means "not".

All of these are different ways to denote the complement of A. Now that we know what the complement is, let's dive deeper into our dice roll example here. So, in this example, we're asked, when rolling a six-sided die, what is the probability that we will roll a four? If we refer to this event as A, the probability of A is equal to the number of outcomes that include that event. So in this case, there's only one way I could roll a four, divided by the number of total possible outcomes.

So since this is a six-sided die, all my total outcomes are six. So the probability of A is 16. But what about the probability that we will not roll a four or the probability of the complement of A? Well, I know that I have five possible outcomes that I would not roll a four because I could roll a one, two, three, five, or six. Here I take all of the outcomes that include that event, not rolling a four, and divide it by the number of total possible outcomes, in this case, still six.

Now looking at these, if I were to take the probability of A and the probability of its complement and add them together, I see that I get six over six, which is just one. Now this makes sense, right, because we've covered 100% of the possibilities of rolling a six-sided die, rolling a one, two, three, five, six, or a four. So it makes sense that the total probability of all possible events is simply one. This is always going to be true. The probability of some event plus the probability of its complement is going to be equal to one.

We can use this formula over here to more easily calculate the probability of something not happening by rearranging a little bit here. So, if I were to subtract the probability of A from both sides here, it will cancel on the left side, leaving me to see that the probability of the complement of A is equal to 1-P(A). So here I see that the probability that something does not happen is simply one minus the probability that it will happen. So now that we know this formula, let's apply it to another example here. In this example, I'm asked when drawing a single card from a standard deck of 52, what is the probability that I will not draw a queen?

Well, instead of trying to find all of the cards that are not a queen, let's just consider all of the cards that are queens. So if I look at the probability of getting a queen, I know that in a standard deck of 52 cards, there are four queens, so I take all of the outcomes that include my event, drawing a queen, and put that over the number of total possible outcomes. In this case, since I have 52 total cards, my total is 52. Then to find the probability of not drawing a queen, I can simply take one minus the probability of drawing that queen, which we just calculated. So we can go ahead and plug in that four over 52 here.

Now I know that one is simply the same thing as 52 over 52, just getting a common denominator here. So if I perform this subtraction, I end up with 4852, which as a decimal is 0.92. So the probability of not drawing a queen is 0.92, and we found that by simply using the probability of drawing a queen without having to count up those 48 cards. So now that we know how to find the probability of something not happening, let's get some more practice. Thanks for watching, and I'll see you in the next one.

When drawing a marble out of a bag of red, green, and yellow marbles 8 times, a red or yellow marble is drawn 6 times. What is the probability of drawing a green marble?

• A. $0.025$
• B. $0.125$
• C. $0.25$
• D. $0.75$

Answer: (C)

A weatherman states that the probability that it will rain tomorrow is 10%, or 0.1, & the probability that it will snow is 25%, or 0.25. What is the probability that it will not rain or snow?

• A. $0.35$
• B. $0.65$
• C. $0.75$
• D. $0.90$

Answer: (A)

Here’s what students ask on this topic:

What is the complement rule in probability?

The complement rule in probability is a fundamental principle that states the sum of the probabilities of an event and its complement equals one. Mathematically, it is expressed as $P_{\neg }=1-P_A$. This means that if you know the probability of an event occurring, you can easily find the probability of it not occurring by subtracting the event's probability from one. For example, if the probability of rolling a four on a six-sided die is $\frac{1}{6}$, the probability of not rolling a four is $\frac{5}{6}$.

How do you calculate the probability of an event not happening?

To calculate the probability of an event not happening, known as the complement, you use the formula $P_{\neg }=1-P_A$. This formula states that the probability of the complement of an event is equal to one minus the probability of the event itself. For instance, if the probability of drawing a queen from a standard deck of 52 cards is $\frac{4}{52}$, the probability of not drawing a queen is $\frac{48}{52}$.

Why is the sum of the probabilities of an event and its complement always equal to one?

The sum of the probabilities of an event and its complement is always equal to one because they encompass all possible outcomes of a scenario. In probability, the total probability of all possible outcomes must equal one, representing certainty that one of the outcomes will occur. For example, when rolling a six-sided die, the probability of rolling a four plus the probability of not rolling a four covers all possible outcomes (rolling a one, two, three, five, or six). Mathematically, this is expressed as $P_A+P_{\neg }=1$.

How can the complement rule simplify probability calculations?

The complement rule simplifies probability calculations by allowing you to find the probability of an event not occurring without having to count all possible outcomes manually. Instead of calculating the probability of each outcome that does not include the event, you can simply subtract the probability of the event from one. This is particularly useful in complex scenarios where counting all non-event outcomes is cumbersome. For example, in a deck of 52 cards, finding the probability of not drawing a queen is easier by using the complement rule: $P_{\neg }=1-\frac{4}{52}$.

What are some common notations for representing the complement of an event?

There are several common notations for representing the complement of an event in probability. These include using an apostrophe after the event symbol, a line over the event symbol, or a special symbol in front of the event symbol that indicates 'not'. For example, if the event is denoted as 'A', its complement can be represented as 'A′', 'Ā', or '¬A'. These notations help distinguish between the event itself and its complement, making it easier to apply the complement rule in probability calculations.