Question

1. how can you determine whether two events are independent or dependent?

1. determine whether the events are independent or dependent. explain your reasoning.

a. you roll a 4 on a six - sided die and spin red on a spinner.

b. your teacher chooses a student to lead a group, chooses another student to a second group, and chooses a third student to lead a third group.

Explanation:

Question 4

To determine if two events are independent or dependent, we use the definition of independent events: two events \( A \) and \( B \) are independent if the occurrence of one event does not affect the probability of the other event occurring. Mathematically, this is expressed as \( P(A \cap B) = P(A) \times P(B) \). For dependent events, the occurrence of one event affects the probability of the other event, so \( P(A \cap B)
eq P(A) \times P(B) \) (or we can reason about whether the outcome of one event changes the possible outcomes or probabilities of the other event).

For the events "rolling a 4 on a six - sided die" and "spinning red on a spinner", we check if the outcome of one event affects the outcome of the other. The result of rolling a die (getting a 4 or any other number) has no impact on the result of spinning a spinner (the probability of spinning red remains the same regardless of what we rolled on the die). So, by the definition of independent events, these two events are independent.

When the teacher chooses a student to lead a group, then chooses another student for a second group, and then a third for a third group, the first choice affects the subsequent choices. After choosing the first student, there are fewer students left to choose from for the second group, and after choosing the second student, there are even fewer for the third group. So the outcome of each previous choice (of a group leader) affects the probability of the next choice (since the pool of available students is reduced). Thus, these events are dependent.

Answer:

Two events are independent if the occurrence of one event does not affect the probability of the other event occurring (mathematically, \( P(A \cap B) = P(A) \times P(B) \)). They are dependent if the occurrence of one event affects the probability of the other event (so \( P(A \cap B)
eq P(A) \times P(B) \) or the outcome of one event changes the possible outcomes/probabilities of the other).

Question 5a