Determine which of the following are examples of independent events.

Here are the examples of independent events from the list:

1.

• a. Rolling a 5 on one die and rolling a 5 on a second die. The outcome of one die does not influence the outcome of the other die. $$\boxed{\text{Independent}}$$
• b. Randomly picking a cookie from the cookie jar and picking a jack from a deck of cards. These are two entirely separate events that do not affect each other. $$\boxed{\text{Independent}}$$
• c. Winning a hockey game and scoring a goal. Scoring a goal directly contributes to and influences the probability of winning a hockey game. These are dependent events.

2.

• a. Choosing an 8 from a deck of cards, replacing it, and choosing a face card. Since the card is replaced, the deck returns to its original state, and the first choice does not affect the probability of the second choice. $$\boxed{\text{Independent}}$$
• b. Going to the beach and bringing an umbrella. Bringing an umbrella is often related to or influenced by the decision to go to the beach (e.g., for shade or expected weather). These are dependent events.
• c. Getting gasoline for your car and getting diesel fuel for your car. A car typically uses either gasoline or diesel, but not both. If you get one, you cannot get the other for the same car at the same time. These are mutually exclusive and thus dependent events.