What are Sample Spaces and Events?

Sample Spaces

What is a Sample Space?

The sample space of an experiment is the population of all unique possible results that could occur in that experiment. It is typically denoted by $S$

What is an Outcome?

An outcome is a single possible result of an experiment. Similar to data, we often $x$ to denote an outcome.

Example 1

Consider rolling a fair six-sided die. What is the sample space for this experiment?

Solution

The sample space consists of all possible outcomes when rolling a six-sided die. \[ S = \{1, 2, 3, 4, 5, 6\} \]

$$\tag*{$\blacksquare$}$$

Example 2

Consider rolling two fair four-sided dice (2d4). What is the sample space for this experiment?

Solution

Each die has four possible outcomes: $ \{1, 2, 3, 4\} $. Since two dice are rolled, each result consists of two digits, where:

The first digit represents the result of the first die.
The second digit represents the result of the second die.

These are not single numbers but rather a shorthand notation for listing two separate results. Instead of writing the outcome as $ (2,3) $ or $ 2,3 $, we omit unnecessary commas and parentheses to focus on the values and their positions.

So, when you see $ 23 $ in this example, think of rolling a $ 2 $ on the first die and a $ 3 $ on the second die, rather than interpreting it as the number twenty-three.

The sample space consists of all possible outcomes:

S = { 11,  12,  13,  14,  
          21,  22,  23,  24,  
          31,  32,  33,  34,  
          41,  42,  43,  44 }

$$\tag*{$\blacksquare$}$$

Example 3

Consider flipping a fair coin 4 times. Each flip results in either Heads (H) or Tails (T). The outcome of the experiment is a sequence of four letters representing the results of each flip. For example, the outcome THHT represents getting tails on the first flip, heads on the second and third flips, and tails on the last flip.

Solution

Each flip has two possible outcomes: $ \{H, T\} $. Since the coin is flipped 4 times, each result consists of a sequence of four letters, where:

The first letter represents the result of the first flip.
The second letter represents the result of the second flip.
The third letter represents the result of the third flip.
The fourth letter represents the result of the fourth flip.

Similar to the dicerolling example, we think of these sequence of letters as an list representing the result of each flip. Instead of writing the outcome as $ (T, H, H, T) $ or $ T, H, H, T $, we omit unnecessary commas and parentheses to focus on the sequence and its position.

So, when you see $ THHT $ in this example, think of flipping Tails first, then Heads twice, then Tails last.

The sample space consists of all possible sequences of four coin flips:

S = { HHHH,  HHHT,  HHTH,  HHTT,
          HTHH,  HTHT,  HTTH,  HTTT,
          THHH,  THHT,  THTH,  THTT,
          TTHH,  TTHT,  TTTH,  TTTT }

$$\tag*{$\blacksquare$}$$

Events

What is an Event?

An event is a subset is any collection of results from sample space that satisfy a given condition.

Example 4

Consider rolling a fair six-sided die. The sample space is:

$ S = \{1, 2, 3, 4, 5, 6\} $

List the outcomes for the event of rolling an even number.

Solution

The event of rolling an even number consists of the outcomes where the result is divisible by 2:

$ E = \{2, 4, 6\} $

$$\tag*{$\blacksquare$}$$

Example 5

Consider rolling two fair four-sided dice (2d4). The sample space consists of all two-digit sequences where each digit represents a separate die roll:

S = { 11  12  13  14  
          21  22  23  24  
          31  32  33  34  
          41  42  43  44 }

List the outcomes for the event of rolling a total sum of 5.

Solution

The event consists of all outcomes where the sum of the two dice equals 5:

$ E = \{14, 23, 32, 41\} $

$$\tag*{$\blacksquare$}$$

Example 6

Consider flipping a fair coin 4 times. The sample space consists of all possible sequences of four flips:

S = { HHHH  HHHT  HHTH  HHTT
          HTHH  HTHT  HTTH  HTTT
          THHH  THHT  THTH  THTT
          TTHH  TTHT  TTTH  TTTT }

List the outcomes for the event of getting two or more heads.

Solution

The event consists of all outcomes where at least two of the four flips are heads:

E = { HHHH  HHHT  HHTH  HHTT
          HTHH  HTHT  HTTH  
          THHH  THHT  THTH  
          TTHH }

$$\tag*{$\blacksquare$}$$

The Number of Outcomes

How Do I Denote the Number of Outcomes in a Sample Space or Event?

The notation $ n(A) $ represents the number of elements in a set $ A $.

When applied to a sample space $ S $, $ n(S) $ gives the total number of possible outcomes in the experiment.
Similarly, for an event $ E $, $ n(E) $ represents the number of outcomes that satisfy the event's condition.

Example 7

Consider rolling a fair six-sided die. The sample space is: $ S = \{1, 2, 3, 4, 5, 6\} $. The event of rolling an even number is: $ E = \{2, 4, 6\} $. Compute $ n(S) $ and $ n(E) $.

Solution

$ n(S) = 6 $, since there are 6 possible outcomes.
$ n(E) = 3 $, since there are 3 even numbers in the sample space.

$$\tag*{$\blacksquare$}$$

Example 8

Consider rolling two fair four-sided dice (2d4). The sample space consists of all two-digit sequences where each digit represents a separate die roll:

S = { 11  12  13  14  
          21  22  23  24  
          31  32  33  34  
          41  42  43  44 }

The event of rolling a total sum of 5 is:

E = { 14  23  32  41 }

Compute $ n(S) $ and $ n(E) $.

Solution

$ n(S) = 16 $, since there are 16 possible rolls.
$ n(E) = 4 $, since there are 4 outcomes where the sum is 5.

$$\tag*{$\blacksquare$}$$

Example 9

Consider flipping a fair coin 4 times. The sample space consists of all possible sequences of four flips:

S = { HHHH  HHHT  HHTH  HHTT
          HTHH  HTHT  HTTH  HTTT
          THHH  THHT  THTH  THTT
          TTHH  TTHT  TTTH  TTTT }

The event of getting two or more heads is:

E = { HHHH  HHHT  HHTH  HHTT
          HTHH  HTHT  HTTH  THHH  
      THHT  THTH  TTHH }

Compute $ n(S) $ and $ n(E) $.

Solution

$ n(S) = 16 $, since there are 16 possible flip sequences.
$ n(E) = 10 $, since there are 10 sequences that contain at least 2 heads.

Theoretical Probablity

What is Theoretical Probability?

The theoretical probability of an event $ E $ occurring is given by the ratio:

\[ P(E) = \dfrac{n(E)}{n(S)} \]

where:

$ n(E) $ is the number of outcomes in the event.
$ n(S) $ is the total number of outcomes in the sample space.

Theoretical probability assumes that all outcomes in the sample space are equally likely.

Example 10

Consider rolling a fair six-sided die. The sample space is:

$ S = \{1, 2, 3, 4, 5, 6\} $

The event of rolling an even number is:

$ E = \{2, 4, 6\} $

Compute $ P(E) $.

Solution

$ n(S) = 6 $, since there are 6 possible outcomes.
$ n(E) = 3 $, since there are 3 even numbers.

Using the formula for theoretical probability:

\[ P(E) = \dfrac{n(E)}{n(S)} = \dfrac{{3}}{{6}} = \dfrac{{1}}{{2}} \]

$$\tag*{$\blacksquare$}$$

Example 11

Consider rolling two fair four-sided dice (2d4). The sample space consists of all two-digit sequences where each digit represents a separate die roll:

S = { 11  12  13  14  
          21  22  23  24  
          31  32  33  34  
          41  42  43  44 }

The event of rolling a total sum of 5 is:

E = { 14  23  32  41 }

Compute $ P(E) $.

Solution

$ n(S) = 16 $, since there are $ 4 \times 4 = 16 $ possible rolls.
$ n(E) = 4 $, since there are 4 outcomes where the sum is 5.

Using the formula for theoretical probability:

\[ P(E) = \dfrac{n(E)}{n(S)} = \dfrac{{4}}{{16}} = \dfrac{{1}}{{4}} \]

$$\tag*{$\blacksquare$}$$

Example 12

Consider flipping a fair coin 4 times. The sample space consists of all possible sequences of four flips:

S = { HHHH  HHHT  HHTH  HHTT
          HTHH  HTHT  HTTH  HTTT
          THHH  THHT  THTH  THTT
          TTHH  TTHT  TTTH  TTTT }

The event of getting two or more heads is:

E = { HHHH  HHHT  HHTH  HHTT
          HTHH  HTHT  HTTH  THHH  \
      THHT  THTH  TTHH }

Compute $ P(E) $.

Solution

$ n(S) = 16 $, since there are $ 2^4 = 16 $ possible flip sequences.
$ n(E) = 10 $, since there are 10 sequences that contain at least 2 heads.

Using the formula for theoretical probability:

\[ P(E) = \dfrac{n(E)}{n(S)} = \dfrac{{10}}{{16}} = \dfrac{{5}}{{8}} \]

$$\tag*{$\blacksquare$}$$

Putting it all together

Example 13

A nurse is monitoring 3 patients in a clinic for signs of fever (F) or no fever (N). Based on prior data, each patient has a 75% chance of having a fever. The sample space consists of all possible outcomes for the 3 patients.

List the sample space and compute the probability that exactly 2 patients have a fever.

Solution

Each patient can either have a fever (F) or no fever (N). Since there are 3 patients, the sample space consists of all sequences of three letters:

S = { FFF  FFN  FNF  FNN  
          NFF  NFN  NNF  NNN }

Since each patient has one of two possible outcomes, the total number of outcomes in the sample space is:

$ n(S) = 2^3 = 8 $

Now, list the outcomes where exactly 2 patients have a fever:

E = { FFN  FNF  NFF }

Counting these outcomes gives:

$ n(E) = 3 $

Using the probability formula:

\[ P(E) = \dfrac{n(E)}{n(S)} = \dfrac{{3}}{{8}}=0.375 \]

Thus, the probability that exactly 2 patients have a fever is 3/8 or 37.5%.

$$\tag*{$\blacksquare$}$$