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Represent compound events using organized lists

An organized list represents the complete sample space of a compound experiment by systematically recording each possible outcome, such as every ordered pair from two successive choices, without omissions or repetitions. This representation supports finding the probability of an event by comparing favorable outcomes with the total number of equally likely outcomes and clarifies why order may matter; formal permutation and combination formulas or more advanced probability models are outside this scope.

Detailed Explanation: Represent compound events using organized lists

When an experiment has two or more steps, make an organized list to show every possible outcome.

A good method is:

  1. List the possible results of the first step.
  2. For each result, list every possible result of the second step.
  3. Check that no outcomes are missing or repeated.
  4. Count the outcomes that match the event you are interested in.

Example: A fair coin is flipped, and then a fair six-sided number cube is rolled. What is the probability of getting heads and an even number?

The coin can show HH or TT. The number cube can show 1,2,3,4,5,1,2,3,4,5, or 66.

Make an organized list by pairing each coin result with every number:

123456H(H,1)(H,2)(H,3)(H,4)(H,5)(H,6)T(T,1)(T,2)(T,3)(T,4)(T,5)(T,6)\begin{array}{c|cccccc} & 1 & 2 & 3 & 4 & 5 & 6\\ \hline H & (H,1) & (H,2) & (H,3) & (H,4) & (H,5) & (H,6)\\ T & (T,1) & (T,2) & (T,3) & (T,4) & (T,5) & (T,6) \end{array}

There are 1212 total outcomes. Each ordered pair shows the result of the coin first and the number cube second.

The favorable outcomes are the ones with heads and an even number:

(H,2), (H,4), (H,6)(H,2),\ (H,4),\ (H,6)

There are 33 favorable outcomes, so

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

The organized list helps because it shows the complete sample space clearly, with no missing or repeated outcomes.

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