Let's solve this problem step-by-step:
1. Construct the Sample Space
A sequence of three digits where each digit is either 0 or 1 can generate all possible combinations. Here is the complete list (sample space) of all such sequences:
[tex]\[
\{000, 001, 010, 011, 100, 101, 110, 111\}
\][/tex]
This sample space consists of a total of 8 sequences.
2. Count Sequences with Exactly One 0
We need to find the number of sequences that contain exactly one 0. Let’s go through each sequence and count those that fit this condition:
- 000: Contains three 0s
- 001: Contains two 0s
- 010: Contains two 0s
- 011: Contains one 0
- 100: Contains two 0s
- 101: Contains one 0
- 110: Contains one 0
- 111: Contains zero 0s
Sequences that contain exactly one 0 are: 011, 101, and 110.
Therefore, there are 3 such sequences.
3. Calculate the Probability
Probability is calculated by dividing the number of favorable outcomes by the total number of outcomes in the sample space.
[tex]\[
\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{3}{8}
\][/tex]
4. Select the Correct Answer
From the given choices, we seek the one that correctly lists the sample space and the probability:
- a. [tex]$000, 001, 010, 011, 100, 101, 110, 111$[/tex]; the probability is [tex]$\frac{7}{8}$[/tex].
- b. [tex]$001, 011, 101, 111$[/tex]; the probability is [tex]$\frac{2}{8}$[/tex].
- c. [tex]$000, 010, 011, 101, 111$[/tex]; the probability is [tex]$\frac{2}{8} = \frac{1}{4}$[/tex].
- d. [tex]$000, 001, 010, 011, 100, 101, 110, 111$[/tex]; the probability is [tex]$\frac{3}{8}$[/tex].
The correct choice is:
[tex]\[
\boxed{d}
\][/tex]
This solution ensures that we constructed the sample space correctly, counted the sequences with exactly one 0 carefully, and calculated the probability accurately.
