What is meant by the sample space for a random experiment?
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The sample space for a random experiment refers to the set of all possible outcomes or results that can occur in that experiment. It is denoted by the symbol Ω.
To understand the concept of a sample space, let's take an example. Suppose we have a fair six-sided die and we want to determine the sample space for rolling the die. The possible outcomes when rolling a die are the numbers 1, 2, 3, 4, 5, and 6. Therefore, the sample space for this random experiment is {1, 2, 3, 4, 5, 6}.
The sample space can be finite, infinite, or even uncountably infinite, depending on the nature of the experiment. For instance, when flipping a coin, the sample space consists of two outcomes: heads or tails. In contrast, the sample space for measuring the height of individuals can be infinitely large, as there is no upper limit to the heights people can have.
The importance of identifying the sample space lies in its ability to help us calculate probabilities. By knowing all the possible outcomes, we can determine the likelihood of a particular event occurring by comparing the favorable outcomes to the total number of outcomes in the sample space.