https://file.aiquickdraw.com/mathbot/file/1727291504830i6z2llja.jpg

Let's analyze the two parts of the problem displayed in the image.

### (a) **Independent Events**:  
The statement is:  
*"Picking a Jack and picking a ♦ are independent events."*

- **Independent events**: Two events are independent if the occurrence of one event does not affect the probability of the other event.
  
Here, "picking a Jack" refers to selecting any card that is a Jack, and "picking a ♦" refers to selecting any card that is a diamond suit.

- There are **4 Jacks** (one in each suit: ♥, ♦, ♣, ♠).
- There are **4 Diamonds** (one for each rank: Jack, Queen, King, Ace, 10).
  
The event of picking a Jack is not independent of picking a ♦ because one of the Jacks is a ♦ (the Jack of Diamonds). Thus, these events are **dependent**.

So, the correct answer for **(a)** is **A**:  
*The statement is false. Picking a Jack and picking a ♦ are dependent events.*

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### (b) **Mutually Exclusive Events**:  
The statement is:  
*"Picking an Ace and picking a ♠ are mutually exclusive events."*

- **Mutually exclusive events**: Two events are mutually exclusive if they cannot happen at the same time.

There is an **Ace of Spades (♠)** in the deck, meaning that it is possible to pick a card that is both an Ace and a ♠. Since both events can happen simultaneously, they are **not mutually exclusive**.

So, the correct answer for **(b)** is **A**:  
*The statement is false. Picking an Ace and picking a ♠ are not mutually exclusive events.*

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Would you like further clarification or more details?  
Here are some related questions for further exploration:

1. What is the difference between independent and dependent events?
2. Can mutually exclusive events be dependent?
3. How does the probability of dependent events change compared to independent events?
4. What other conditions can make two events mutually exclusive?
5. How would the analysis change with a full deck of 52 cards?

**Tip**: When analyzing events, always consider whether the occurrence of one event changes the likelihood of the other happening. This helps in distinguishing between dependent and independent events.

An abbreviated deck of cards consists of four Jacks, four Queens, four Kings, four Aces, and four 10s. For each type of card, there is a Heart, a Diamond, a Club, and a Spade. You are to randomly pick a single card from this deck.

(a) Consider the statement: 'Picking a Jack and picking a Diamond are independent events.' Is this statement true or false?

(b) Consider the statement: 'Picking an Ace and picking a Spade are mutually exclusive events.' Is this statement true or false?

Explore probability concepts with this example using an abbreviated deck of cards. Understand independent and dependent events as well as mutually exclusive events through simple scenarios. Suitable for Grades 9-12.

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