"Classifying Events as Independent or Dependent In Exercises 9-14, determine whether the events are independent or dependent. Explain your reasoning.
10. A father having hazel eyes and a daughter having hazel eyes"

Verified step by step guidance

1. Step 1: Understand the concept of independent and dependent events. Independent events are those where the occurrence of one event does not affect the probability of the other event. Dependent events are those where the occurrence of one event does affect the probability of the other event.
2. Step 2: Analyze the relationship between the father having hazel eyes and the daughter having hazel eyes. Consider whether the genetic inheritance of eye color plays a role in determining the probability of the daughter having hazel eyes.
3. Step 3: Recall that eye color is influenced by genetics. A father having hazel eyes increases the likelihood of the daughter inheriting hazel eyes due to shared genetic traits. This suggests a dependency between the two events.
4. Step 4: Formulate the reasoning: Since the probability of the daughter having hazel eyes is affected by the father’s eye color (due to genetic inheritance), these events are dependent.
5. Step 5: Conclude that the events are dependent and explain that the dependency arises from the genetic relationship between the father and daughter, which influences the likelihood of shared traits like eye color.

Key Concepts

Independent Events

Independent events are those whose outcomes do not affect each other. In probability, two events A and B are independent if the occurrence of A does not change the probability of B occurring, and vice versa. For example, flipping a coin and rolling a die are independent events because the result of one does not influence the other.

Dependent Events

Dependent events are those where the outcome of one event affects the outcome of another. In probability, two events A and B are dependent if the occurrence of A changes the probability of B occurring. For instance, drawing cards from a deck without replacement creates dependent events, as the first draw alters the composition of the deck for the second draw.

Genetic Inheritance

Genetic inheritance refers to the way traits and characteristics are passed from parents to offspring through genes. In the context of the question, the eye color of a father and daughter can be influenced by their genetic makeup, which may indicate a dependent relationship. The probability of the daughter having hazel eyes is affected by the father's eye color, illustrating how genetic traits can be interrelated.