Events as Subsets of a Sample Space: Definition & Example

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  • 0:02 Events, Subsets and…
  • 0:41 Definitions
  • 2:43 Examples
  • 3:56 Lesson Summary
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Lesson Transcript
Instructor: Cathryn Jackson

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

Probability can get very confusing at times. You will find that some words, such as events and subsets, are often referring to the same concept depending on the experiment. Use this lesson to understand the concept of events as subsets.

Events & Sample Spaces

Bianca, Megan and Nick are sitting at their favorite Italian restaurant. It's lunchtime, so the waiter hands them each a lunch menu. The menu is comprised of four different items: lasagna, spaghetti, fettuccine Alfredo or pizza. They can also order a sampler, which is a small portion of each of the items on the menu. There are a lot of different combinations that these friends can have as a meal. In statistics, you will find experiments that have many possible outcomes and combinations. What do we call these experiments and outcomes? Some terms that you will come across are subsets, events and sample spaces.

Defining Events, Subsets & Sample Spaces

When you conduct an experiment, you are observing certain outcomes. For example, you may be conducting an experiment on flipping a coin. The possible outcomes for flipping a coin are heads or tails. If you were rolling a six-sided die, then the possible outcomes would be 1, 2, 3, 4, 5 or 6. There are no other possible outcomes. We call these possible outcomes sample spaces. A sample space is a set (S) of a random experiment that includes all possible outcomes of the experiment.

In our opening scenario, one of the options on the menu was a sampler. Let's say that the sampler is a small portion of everything else that's on the menu. You can think about sample spaces as the sampler on a menu; it gives you an idea or picture of all of the possible outcomes in the experiment. For example, if Megan ordered the sampler, she knows she's not going to get tacos as part of her meal. Tacos are not part of the sample space in this case.

Now that we know about sample space, what about the actual results of the experiment? You will hear two words in statistics: events and subsets. Before we define these things, you must remember something very important: events are subsets, and subsets are events. An event is a possible outcome of an experiment. And a subset is an event of a sample space. Therefore, if Bianca ordered a pizza, that would be an event, or subset, of the sample space.

When you think of subsets, think of what that word means. Look at the prefix 'sub-'. Think of the words you know that include 'sub-' as a prefix: subway, submarine, subordinate, subheading. In this case, sub- means below or beneath. A subheading for an essay is something that is beneath and part of the main heading. Think of the subset as being beneath, or a part of, the sample space.

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