Answer
Choosing two socks from a drawer affects the probability of the second pick, making the selections dependent events.
Solution
Choosing two socks from a drawer containing blue, red, and green socks is an example of **dependent events** because the outcome of the second selection is influenced by the outcome of the first selection. Here's a detailed explanation:
### **Understanding Dependent Events**
**Dependent events** are events where the outcome or occurrence of the first event affects the outcome or probability of the second event. In other words, the probability of the second event **depends** on what happened in the first event.
### **Applying This to Selecting Socks**
1. **First Sock Selection:**
- Suppose you have a drawer with a mix of blue, red, and green socks.
- When you pick the first sock, you remove it from the drawer, which changes the total number of socks and potentially the number of socks of each color remaining.
2. **Impact on Second Sock Selection:**
- The color and number of socks now left in the drawer depend on what you picked first.
- For example:
- **If you picked a blue sock first:** There are now fewer blue socks in the drawer, which changes the probability of picking another blue sock on the second draw.
- **If you picked a red sock first:** The ratio of red to other colored socks has changed, affecting the likelihood of picking red, blue, or green on the second draw.
- **If you picked a green sock first:** Similarly, the probabilities for the second sock adjust based on the remaining green socks.
3. **Probabilistic Dependence:**
- Mathematically, if the first event changes the sample space or alters the probabilities of outcomes for the second event, the events are dependent.
- In the sock example, the probability of drawing a certain color on the second draw is directly influenced by what was drawn first. This linkage means the two draws are not independent of each other.
### **Contrast with Independent Events**
- **Independent events** are those where the outcome of one event does not affect the outcome of another. For example, flipping a coin twice: the result of the first flip doesn’t influence the second flip.
- In the sock selection scenario, since removing a sock alters the composition and probabilities for the next pick, they are **not independent**.
### **Real-World Implications**
Understanding that these events are dependent is crucial in probability calculations. It requires adjusting the probabilities after each draw, typically using conditional probability formulas to account for the changing situation.
### **Conclusion**
Selecting two socks from a drawer without replacing the first one makes the two selections **dependent events** because the first selection influences the probabilities and outcomes of the second selection.
Reviewed and approved by the UpStudy tutoring team
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