Question

video lesson 5.2 example 1 & 2 - determining whether events are independent

1. a bag contains six pieces of paper, numbered 1 through 6. a student randomly selects a piece of paper, does not replace it, and randomly selects another piece of paper. use a sample space to determine whether randomly selecting an even number first and randomly selecting a 4 second are independent events.

Explanation:

Step1: Define Events

Let \( A \) be the event of selecting an even number first, and \( B \) be the event of selecting a 4 second. The numbers are 1, 2, 3, 4, 5, 6. Even numbers: 2, 4, 6. So \( n(A) \) (first selection even) and then second selection (without replacement).

Step2: Sample Space for First Selection

First, list all possible first selections:

• If first is 2: second can be 1,3,4,5,6 (5 outcomes)
• If first is 4: second can be 1,2,3,5,6 (5 outcomes)
• If first is 6: second can be 1,2,3,4,5 (5 outcomes)

Total outcomes for two - step selection: \( 6\times5 = 30 \) (since no replacement).

Step3: Calculate \( P(A) \)

\( P(A)=\frac{\text{Number of ways to choose even first}}{\text{Total number of first - selection ways}}=\frac{3}{6}=\frac{1}{2} \).

Step4: Calculate \( P(B) \)

To find \( P(B) \), total number of ways to select 4 second:

• If first is not 4: first can be 1,2,3,5,6 (5 numbers), then second is 4. So 5 outcomes.
• If first is 4: second can't be 4 (no replacement). So total outcomes with 4 second: 5.

\( P(B)=\frac{5}{30}=\frac{1}{6} \).

Step5: Calculate \( P(A\cap B) \)

\( A\cap B \) is selecting even first and 4 second. Even first numbers: 2,4,6.

• If first is 2: second is 4 (1 outcome: (2,4))
• If first is 4: second can't be 4 (no replacement)
• If first is 6: second is 4 (1 outcome: (6,4))

So \( n(A\cap B)=2 \), \( P(A\cap B)=\frac{2}{30}=\frac{1}{15} \).

Step6: Check Independence

Two events \( A \) and \( B \) are independent if \( P(A\cap B)=P(A)\times P(B) \).
\( P(A)\times P(B)=\frac{1}{2}\times\frac{1}{6}=\frac{1}{12} \), and \( P(A\cap B)=\frac{1}{15} \). Since \( \frac{1}{15}
eq\frac{1}{12} \), the events are not independent.

Answer:

The events of randomly selecting an even number first and randomly selecting a 4 second are not independent.