Represent sample spaces using organized lists

An organized sample space lists every possible outcome of a chance experiment exactly once in a systematic order, including compound outcomes such as ordered pairs from two successive choices; tables or simple tree diagrams may clarify the structure. It distinguishes individual outcomes from events (sets of outcomes), prevents omissions and repetitions, and supports finding probabilities by counting favorable outcomes among all equally likely outcomes, without extending to formal permutations, combinations, or generalized counting formulas.

Detailed Explanation: Represent sample spaces using organized lists

An outcome is one possible result of a chance experiment. A sample space is a list of all possible outcomes.

To make an organized list:

Decide what each part of an outcome records.
Keep the order clear.
List every possibility for the first part.
For each first possibility, list every possibility for the next part.
Check that no outcome is missing or repeated.

Example: A fair coin is tossed twice. What is the sample space?

The outcome must show the result of the first toss and the second toss. Use $H$ for heads and $T$ for tails.

Start with the first toss being $H$ :

$(HH)$ : heads, then heads
$(HT)$ : heads, then tails

Now start with the first toss being $T$ :

$(TH)$ : tails, then heads
$(TT)$ : tails, then tails

So the organized sample space is

\{HH,\ HT,\ TH,\ TT\}.

There are $4$ possible outcomes. Notice that $(HT)$ and $(TH)$ are different outcomes because the order is different.

If the event is “getting exactly one head,” the favorable outcomes are

\{HT,\ TH\}.

There are $2$ favorable outcomes out of $4$ equally likely outcomes, so

P(\text{exactly one head})=\frac{2}{4}=\frac{1}{2}.

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Probability Sample Space - Definition to Sample Space List

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Represent sample spaces using organized lists

Detailed Explanation: Represent sample spaces using organized lists

Learn by doing: Represent sample spaces using organized lists

Practice with unlimited practice problems

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Probability Sample Space - Definition to Sample Space List