Question

consider a single spin on the spinner shown below.

which events are mutually exclusive? check all that apply.

landing on an unshaded portion and landing on 2
landing on a shaded portion and landing on 3
landing on a shaded portion and landing on an even number
landing on an unshaded portion and landing on a number greater than 3
landing on a shaded portion and landing on an unshaded portion

Explanation:

Define the sample space and outcomes

The spinner is divided into four equal sectors labeled 1, 2, 3, and 4.

• Shaded portions: sectors 1 and 2.
• Unshaded portions: sectors 3 and 4.
• Even numbers: 2 and 4.
• Odd numbers: 1 and 3.

Define mutually exclusive events

Using the Theoretical Probability and Sample Space knowledge points, two events are mutually exclusive if they cannot occur at the same time. This means their intersection is empty:

$$ P(A \cap B) = 0 $$

Analyze each option

• Option 1: landing on an unshaded portion (3, 4) and landing on 2.

The intersection is empty because 2 is in a shaded portion. These events are mutually exclusive.

• Option 2: landing on a shaded portion (1, 2) and landing on 3.

The intersection is empty because 3 is in an unshaded portion. These events are mutually exclusive.

• Option 3: landing on a shaded portion (1, 2) and landing on an even number (2, 4).

The outcome 2 is in both events. These events are not mutually exclusive.

• Option 4: landing on an unshaded portion (3, 4) and landing on a number greater than 3 (4).

The outcome 4 is in both events. These events are not mutually exclusive.

• Option 5: landing on a shaded portion (1, 2) and landing on an unshaded portion (3, 4).

The intersection is empty. These events are mutually exclusive.

Answer:

• landing on an unshaded portion and landing on 2 (Correct answer)
• landing on a shaded portion and landing on 3 (Correct answer)
• landing on a shaded portion and landing on an even number
• landing on an unshaded portion and landing on a number greater than 3
• landing on a shaded portion and landing on an unshaded portion (Correct answer)