Use organized lists for sample spaces

An organized list, table, or tree-like representation identifies every possible outcome of a finite chance experiment, such as two coin tosses or selecting items in sequence, without omissions or repetitions. The learner distinguishes individual outcomes from events (sets of outcomes) and can use a complete sample space to determine probabilities when outcomes are equally likely, recognizing that an incomplete list produces incorrect results. The focus is on small, finite experiments rather than general counting formulas or more advanced combinatorial methods.

Detailed Explanation: Use organized lists for sample spaces

To use an organized list, write down every possible outcome of the chance experiment exactly once.

Example: Toss a coin twice

Find the probability of getting exactly one head.

Step 1: List the outcomes in order

For each toss, the result can be Heads ( $H$ ) or Tails ( $T$ ).

Make an organized list:

First toss	Second toss	Outcome
$H$	$H$	$HH$
$H$	$T$	$HT$
$T$	$H$	$TH$
$T$	$T$	$TT$

The sample space is

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

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

Step 2: Identify the event

The event is “exactly one head.” The outcomes that have exactly one $H$ are

\{HT,\ TH\}.

This event has $2$ outcomes.

Step 3: Find the probability

All four outcomes are equally likely, so

P(\text{exactly one head}) = \frac{\text{number of favorable outcomes}}{\text{number of possible outcomes}} = \frac{2}{4} = \frac{1}{2}.

The probability of getting exactly one head is

\boxed{\frac{1}{2}}.

An organized list helps prevent missing an outcome or counting one more than once.

Learn by doing: Use organized lists for sample spaces

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Probability Sample Space - Definition to Sample Space List

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