A Drawer Contains One Pair Of Brown Socks And One Pair Of White Socks. The Table Shows The Possible Outcomes, Or Sample Space, For Choosing A Sock, Replacing It, And Then Choosing Another Sock.$\[ \begin{tabular}{|c|c|c|c|c|c|} \hline
**A Drawer Contains One Pair of Brown Socks and One Pair of White Socks: Understanding the Sample Space**
What is the Sample Space?
The sample space is the set of all possible outcomes or results of an experiment. In this case, the experiment is choosing a sock, replacing it, and then choosing another sock. The sample space is a table that shows all the possible combinations of socks that can be chosen.
Understanding the Table
| Outcome 1 | Outcome 2 | Description |
|---|---|---|
| Brown | Brown | Choosing a brown sock and then choosing another brown sock |
| Brown | White | Choosing a brown sock and then choosing a white sock |
| White | Brown | Choosing a white sock and then choosing a brown sock |
| White | White | Choosing a white sock and then choosing another white sock |
Q&A: Understanding the Sample Space
Q: What is the probability of choosing a brown sock first and then a white sock?
A: The probability of choosing a brown sock first is 1/2, since there are two socks in the drawer and one of them is brown. The probability of choosing a white sock second is also 1/2, since the first sock has been replaced and there are again two socks in the drawer, one of which is white. Therefore, the probability of choosing a brown sock first and then a white sock is (1/2) × (1/2) = 1/4.
Q: What is the probability of choosing a white sock first and then a brown sock?
A: The probability of choosing a white sock first is 1/2, since there are two socks in the drawer and one of them is white. The probability of choosing a brown sock second is also 1/2, since the first sock has been replaced and there are again two socks in the drawer, one of which is brown. Therefore, the probability of choosing a white sock first and then a brown sock is (1/2) × (1/2) = 1/4.
Q: What is the probability of choosing two brown socks?
A: The probability of choosing a brown sock first is 1/2, since there are two socks in the drawer and one of them is brown. The probability of choosing a brown sock second is also 1/2, since the first sock has been replaced and there are again two socks in the drawer, one of which is brown. Therefore, the probability of choosing two brown socks is (1/2) × (1/2) = 1/4.
Q: What is the probability of choosing two white socks?
A: The probability of choosing a white sock first is 1/2, since there are two socks in the drawer and one of them is white. The probability of choosing a white sock second is also 1/2, since the first sock has been replaced and there are again two socks in the drawer, one of which is white. Therefore, the probability of choosing two white socks is (1/2) × (1/2) = 1/4.
Q: What is the probability of choosing a brown sock and then a white sock, or a white sock and then a brown sock?
A: The probability of choosing a brown sock first and then a white sock is 1/4, as calculated above. The probability of choosing a white sock first and then a brown sock is also 1/4, as calculated above. Since these two events are mutually exclusive (i.e., they cannot happen at the same time), we can add their probabilities together to get the total probability: 1/4 + 1/4 = 1/2.
Q: What is the probability of choosing two socks of the same color?
A: The probability of choosing two brown socks is 1/4, as calculated above. The probability of choosing two white socks is also 1/4, as calculated above. Since these two events are mutually exclusive (i.e., they cannot happen at the same time), we can add their probabilities together to get the total probability: 1/4 + 1/4 = 1/2.
Conclusion
In this article, we have explored the concept of a sample space and how it can be used to calculate probabilities. We have used a table to show all the possible combinations of socks that can be chosen, and we have calculated the probabilities of various events, such as choosing a brown sock first and then a white sock, or choosing two socks of the same color. By understanding the sample space and how to calculate probabilities, we can gain a deeper understanding of the world around us and make more informed decisions.