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Represent sample spaces using organized lists

A sample space is the complete set of possible outcomes of a chance experiment, and an organized list records each outcome exactly once in a systematic order or grouping. For simple compound experiments, outcomes may be represented as ordered pairs or other clearly labeled combinations, helping distinguish outcomes from events and avoid omissions, duplication, or treating different orders as identical; counting outcomes supports probability calculations when the outcomes are equally likely.

Detailed Explanation: Represent sample spaces using organized lists

A sample space is the complete list of all possible outcomes of a chance experiment. To organize it:

  1. Identify each part of the experiment.
  2. Choose an order for the parts.
  3. Keep the first part fixed while listing every possibility for the second part.
  4. Repeat for each possibility in the first part.
  5. Check that no outcome is missing or repeated.

Example: A fair coin is flipped and a fair six-sided die is rolled. List the sample space and find the probability of getting heads and an even number.

Use ordered pairs:

(coin result,die result)(\text{coin result},\text{die result})

First list all the outcomes when the coin shows heads:

(H,1),(H,2),(H,3),(H,4),(H,5),(H,6)(H,1),(H,2),(H,3),(H,4),(H,5),(H,6)

Then list all the outcomes when the coin shows tails:

(T,1),(T,2),(T,3),(T,4),(T,5),(T,6)(T,1),(T,2),(T,3),(T,4),(T,5),(T,6)

Therefore, the sample space is

S={(H,1),(H,2),(H,3),(H,4),(H,5),(H,6),(T,1),(T,2),(T,3),(T,4),(T,5),(T,6)}.S=\{(H,1),(H,2),(H,3),(H,4),(H,5),(H,6), (T,1),(T,2),(T,3),(T,4),(T,5),(T,6)\}.

There are 2⋅6=122\cdot 6=12 possible outcomes.

The event “heads and an even number” includes only

(H,2),(H,4),(H,6).(H,2),(H,4),(H,6).

There are 33 favorable outcomes out of 1212 equally likely outcomes, so

P(heads and even)=312=14.P(\text{heads and even})=\frac{3}{12}=\frac14.

The organized order makes it easy to see that every possible outcome has been included exactly once.

Learn by doing: Represent sample spaces using organized lists

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    Probability Sample Space - Definition to Sample Space List

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