Scientists and product testers love to run all kinds of experiments, and sample space becomes a useful concept to understand as you run your own experiments. You'll soon see that while this concept involves subjects like probability, it might be much more straightforward than you think. Let's figure out sample space together:
A sample space is a set of values that represent all of the possible outcomes of an experiment. As you might expect, a sample space for a very complicated experiment can be almost endless. The more variables in an experiment, the more possible outcomes. If you choose a card from a deck, there is a "sample space" of 52 outcomes. But if you flip a coin, the sample space is only 2 possible outcomes. Rolling a six-sided die has a sample space of 6 outcomes. As you can see, the concept is quite straightforward.
To properly represent sample spaces in mathematics, we need to communicate the possible outcomes of an event using a specific notation, rather than just stating the number of outcomes.
For a coin, the sample space includes two possible outcomes: heads and tails. We can represent this sample space as , where and .
For a six-sided die, there are six possible outcomes, each corresponding to the numbers on the faces of the die. We can represent the sample space as .
There are a few other concepts we need to cover before we start running our own experiments:
We know that the sample space for a six-sided die is . But what is the "event" that we get a number less than four?
We would write this as .
What about the event that we get an even number?
We would write this as .
But what might happen if we flipped two coins at the same time? We would have to consider all of the possibilities, and we would write this as:
and then heads, and then tails, and then tails, and then heads.
Although these foundational concepts might seem simple, they provide us with a logical system that helps us map out probability with accuracy. But why might we want to run random experiments? Random experiments help to eliminate potential biases by ensuring that each member of the population has an equal chance of being selected for the sample. This helps to ensure that the sample accurately represents the population as a whole and that any differences between the sample and the population are due to chance, rather than systematic biases.
Common Core: High School - Statistics and Probability Flashcards
Probability Theory Practice Tests
Common Core: High School - Statistics and Probability Diagnostic Tests
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