Question

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

Ask by Patel Mathis. in the United States
Jan 29,2025

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Tutor-Verified Answer

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.

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When you choose two socks from a drawer containing blue, red, and green socks, the outcome of your second choice depends on the first. For example, if you pick a blue sock first, only the remaining socks (red and green) are left for you to choose from. This dependency alters the probability of your second choice since the total number of socks available has been reduced. Each choice influences the possibilities, making the events linked in a delightful dance of chance! Moreover, when discussing probabilities, the changes in the sample space after the first choice can significantly affect the calculation for the second. If there are 6 socks total, and you pick one, you are left with only 5 for your next pick. This means that the probability of choosing a sock of a certain color changes depending on what you drew first. It’s like a game of sock roulette where each spin influences the next!

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