To identify independent events, we look for events where the outcome of one does not affect the outcome of the other. Let's analyze the given pairs:

1. Rolling a 4 on a die and pulling a face card from a deck of cards:

   • These are independent because rolling a die and drawing a card are separate actions, and the outcome of one has no effect on the outcome of the other.

2. Pulling a face card from a deck, pulling a card with a heart on it from a deck:

   • These events are dependent if done without replacement because the deck changes after the first card is drawn. If done with replacement, they would be independent.

3. Pulling a king from a deck of cards on your first try, pulling a king from a deck of cards on your second try (without replacing the first card):

   • These are dependent because without replacing the first card, the total number of cards in the deck is reduced, affecting the probability of drawing a second king.

4. Steph Curry making his first free throw attempt, Steph Curry making his second free throw attempt:

   • These events are dependent because his performance on the first attempt could influence his second attempt (e.g., due to fatigue, confidence, or form).

5. Flipping tails on your first coin toss, flipping tails on your second coin toss:

   • These are independent because each coin toss is separate and unaffected by the previous result.

Independent Events:

• Rolling a 4 on a die and pulling a face card from a deck of cards.
• Flipping tails on your first coin toss, flipping tails on your second coin toss.

Do you want more details or have any questions?

Here are 5 related questions to think about:

1. What is the definition of dependent events in probability?
2. How does replacing or not replacing an item affect the independence of events?
3. What is the probability of rolling a 4 on a die and pulling a face card from a deck?
4. How do you calculate the probability of independent events occurring together?
5. What is the difference between mutually exclusive and independent events?

Tip: To check if events are independent, ask yourself if knowing the outcome of one event changes the probability of the other happening.