Explain why choosing two socks from a drawer containing blue, red, and green socks is an example of dependent events.

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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 probability of the second event is affected by the outcome of the first event. In contrast, **independent events** have outcomes that do not influence each other.

### Applying This to Choosing Socks

Imagine you have a drawer with a mix of blue, red, and green socks. Let's break down the process of selecting two socks:

1. **First Selection:**
   - You reach into the drawer and randomly pick one sock. Suppose you pick a blue sock.
   - After this selection, the composition of the drawer changes. There is now one fewer blue sock, and the total number of socks in the drawer has decreased by one.

2. **Second Selection:**
   - You pick a second sock.
   - The probability of picking a blue, red, or green sock now depends on what you picked first.
   - For example, if you initially picked a blue sock, there are fewer blue socks left, which affects the probability of picking another blue sock compared to the other colors.

### Why They Are Dependent

- **Change in Probabilities:** The first sock you choose alters the number and proportion of each color remaining in the drawer. This change means that the probability of selecting a particular color on the second draw is no longer the same as it was before the first draw.
  
- **Influence on the Second Event:** Because the outcome of the first event (the first sock picked) affects the conditions under which the second event (the second sock picked) occurs, the two events are interdependent.

### Example Scenario

Suppose the drawer has:
- 3 blue socks
- 2 red socks
- 1 green sock

**Total socks initially:** 6

- **Probability of picking a blue sock first:** 3/6 = 1/2

**If a blue sock is picked first:**
- Socks remaining: 2 blue, 2 red, 1 green
- **Probability of picking a blue sock second:** 2/5

**If a red sock is picked first:**
- Socks remaining: 3 blue, 1 red, 1 green
- **Probability of picking a red sock second:** 1/5

Notice how the probability of the second event changes based on the first event's outcome. This interdependence characterizes dependent events.

### Contrast with Independent Events

If the events were independent, the outcome of the first sock selection wouldn't affect the second. For example, flipping a coin twice:
- The first flip resulting in heads doesn't change the probability of the second flip being heads or tails. Each flip is independent.

### Conclusion

Selecting two socks from a drawer where the first selection changes the composition and probabilities for the second selection is a clear case of dependent events. The outcome of the first selection directly influences the probabilities associated with the second selection, illustrating their dependence.
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 depends on the first selection.

Explain why choosing two socks from a drawer containing blue, red, and green socks is an example of dependent events.

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