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

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** In probability, **dependent events** are events where the outcome or occurrence of the first event affects the outcome or probability of the second event. This dependency contrasts with **independent events**, where the outcome of one event has no effect on the other. ### **Applying to the Sock Example** 1. **First Draw:** - When you draw the first sock from the drawer, you reduce the total number of socks available for the second draw. - Suppose the drawer initially has: - 5 blue socks - 3 red socks - 2 green socks - **Total:** 10 socks 2. **Second Draw:** - **Without Replacement:** After drawing the first sock, the total number of socks decreases by one, and the composition of sock colors may also change. - For example, if you first draw a blue sock: - Remaining socks: 4 blue, 3 red, 2 green (total 9 socks) - The probability of drawing a second blue sock is now 4/9, whereas initially it was 5/10. - **With Replacement:** If you were to replace the first sock before drawing the second, the events would be independent because the total number and composition of socks remain unchanged. However, in most practical scenarios like selecting socks from a drawer without replacing them, events are dependent. ### **Why They’re Dependent** - **Changing Probabilities:** The probability of drawing a specific color on the second draw changes based on what was drawn first. Drawing a blue sock first makes it slightly less likely to draw another blue sock if not replaced. - **Altered Sample Space:** The first draw alters the total number of socks and possibly the number of socks of each color, directly affecting the second draw's probabilities. ### **Illustrative Example** Imagine you want to draw two blue socks in a row: - **First Draw:** Probability of drawing a blue sock = 5/10 = 1/2 - **Second Draw:** After drawing one blue sock, remaining blue socks = 4; total socks = 9 - Probability of drawing another blue sock = 4/9 - **Combined Probability:** (1/2) * (4/9) = 2/9 If the events were independent, the second probability would remain 1/2, and the combined probability would be (1/2) * (1/2) = 1/4. However, because the first draw affects the second, the events are dependent. ### **Conclusion** In summary, selecting two socks sequentially from a drawer without replacing the first sock makes the two selection events dependent. The first selection alters the probability landscape for the second selection, demonstrating the essence of dependent events in probability theory.