Question

(1) a sample space is defined as \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_

(2) a sample space is usually presented in \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_ notation.

(3) the sample space for rolling two 6-sided dice has \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_ elements.

(4) when two six-sided dice are rolled, the number of outcomes with a sum of eight is \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_.

(5) the sample space for the vowels chosen from the alphabet has \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_ elements.

(6) construct a tree diagram to find the sample space for tossing a coin four times. write your sample space as a set.

Explanation:

🆕 New Concept Discovered: Sample Space and Outcomes
The set of all possible results of an experiment

Step 1: Define a Sample Space

In probability theory, a sample space is the set of all possible outcomes of a random experiment.

Step 2: Identify Sample Space Notation

A sample space is a collection of distinct outcomes, which is mathematically represented as a set. Therefore, it is usually presented in set notation (using curly braces, e.g., \( S = \{ \dots \} \)).

Step 3: Calculate Elements for Rolling Two Dice

Using the Fundamental Counting Principle, since each 6-sided die has \( 6 \) possible outcomes, rolling two dice yields:

$$ 6 \times 6 = 36 \text{ elements} $$

Step 4: Find Outcomes with a Sum of Eight

We list the pairs \( (d_1, d_2) \) from the sample space of two dice that add up to \( 8 \):

• \( (2, 6) \)
• \( (3, 5) \)
• \( (4, 4) \)
• \( (5, 3) \)
• \( (6, 2) \)

There are \( 5 \) such outcomes.

Step 5: Count Vowels in the Alphabet

The vowels in the English alphabet are \( \{a, e, i, o, u\} \). The number of elements in this sample space is \( 5 \).

Step 6: Construct Tree Diagram and Set for Tossing a Coin Four Times

Each toss has two possible outcomes: Heads (\( \text{H} \)) or Tails (\( \text{T} \)).

A tree diagram branches out at each of the four steps:

1. Toss 1: Branches into \( \text{H} \) and \( \text{T} \).
2. Toss 2: Each previous branch splits into \( \text{H} \) and \( \text{T} \) (4 branches).
3. Toss 3: Each branch splits again (8 branches).
4. Toss 4: Each branch splits a final time (16 branches).

Following every path from the start to the end of the tree yields the \( 16 \) elements of the sample space.

Answer:

(1) A sample space is defined as the set of all possible outcomes of a probability experiment.

(2) A sample space is usually presented in set notation.

(3) The sample space for rolling two 6-sided dice has 36 elements.

(4) When two six-sided dice are rolled, the number of outcomes with a sum of eight is 5.

(5) The sample space for the vowels chosen from the alphabet has 5 elements.

(6)

• Tree Diagram Structure:

                  /--- H --- HHHH
            /--- H
           |      \--- T --- HHHT
     /--- H
    |      |      /--- H --- HHTH
    |       \--- T
    |             \--- T --- HHTT
 ---|
    |             /--- H --- HTHH
    |       /--- H
    |      |      \--- T --- HTHT
     \--- T
           |      /--- H --- HTTH
            \--- T
                  \--- T --- HTTT

                  /--- H --- THHH
            /--- H
           |      \--- T --- THHT
     /--- H
    |      |      /--- H --- THTH
    |       \--- T
    |             \--- T --- THTT
 ---|
    |             /--- H --- TTHH
    |       /--- H
    |      |      \--- T --- TTHT
     \--- T
           |      /--- H --- TTTH
            \--- T
                  \--- T --- TTTT

• Sample Space Set:

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