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 **Dependent events** are events where the outcome or occurrence of the first event affects the outcome or probability of the second event. In contrast, **independent events** have outcomes that do not influence each other. ### Applying to the Sock Example Let's break down the sock selection process to see why these events are dependent: 1. **First Sock Selection:** - Suppose the drawer contains a mix of blue, red, and green socks. - When you draw the first sock, the probability of it being any particular color depends on the initial distribution of colors in the drawer. 2. **Second Sock Selection:** - After removing the first sock, the total number of socks in the drawer decreases by one. - Moreover, the number of socks of the color you just drew also decreases (assuming you drew a specific color). ### Why They Are Dependent - **Change in Probabilities:** - If you drew a blue sock first, there are now fewer blue socks left in the drawer. This changes the probability of drawing a blue sock on the second draw compared to the initial probability before any socks were removed. - Similarly, the probabilities for drawing red or green socks on the second draw are also slightly altered because the total number of socks has decreased. - **Influence of First Outcome:** - The first sock's color directly influences the sample space for the second draw. For example, if you drew a red sock first, the likelihood of drawing another red sock second is now dependent on how many red socks were initially present. ### Numerical Example Consider a drawer with: - 4 blue socks - 3 red socks - 2 green socks - **Total:** 9 socks **First Draw:** - Probability of drawing a blue sock: 4/9 **Second Draw (If a blue sock was drawn first):** - Remaining socks: 8 - Remaining blue socks: 3 - Probability of drawing a blue sock now: 3/8 **Second Draw (If a red sock was drawn first):** - Remaining socks: 8 - Remaining red socks: 2 - Probability of drawing a red sock now: 2/8 As seen, the probability of drawing a second sock of a particular color depends on what was drawn first, illustrating that these two events are **dependent**. ### Contrast with Independent Events If the sock selection were an example of independent events, the outcome of the second draw would remain unchanged regardless of the first draw. However, since removing one sock alters the probabilities of the remaining socks, the events are clearly dependent. ### Conclusion In summary, choosing two socks sequentially from a drawer without replacing the first sock creates a scenario where the outcome of the second draw is affected by the first. This interdependence of events classifies the two sock selections as dependent events.