Question

1. you have three white golf balls and two yellow golf balls in a bag. you randomly select one golf ball to hit now and another golf ball to place in your pocket. use a sample space to determine whether randomly selecting a white golf ball first and then a white golf ball second are independent events.

Explanation:

Step1: Define Events

Let \( A \) be the event of selecting a white golf ball first, and \( B \) be the event of selecting a white golf ball second. Total golf balls initially: \( 3 + 2 = 5 \).

Step2: Calculate \( P(A) \)

\( P(A)=\frac{\text{Number of white balls}}{\text{Total balls}}=\frac{3}{5} \).

Step3: Calculate \( P(B|A) \)

After selecting a white ball first, there are \( 3 - 1 = 2 \) white balls and \( 5 - 1 = 4 \) total balls left. So \( P(B|A)=\frac{2}{4}=\frac{1}{2} \).

Step4: Calculate \( P(B) \) (without considering \( A \))

\( P(B)=\frac{3}{5} \) (same as \( P(A) \) initially, but we check if \( P(B|A)=P(B) \)).

Step5: Compare \( P(B|A) \) and \( P(B) \)

\( P(B|A)=\frac{1}{2}=0.5 \), \( P(B)=\frac{3}{5}=0.6 \). Since \( 0.5
eq0.6 \), the events are dependent. Also, using the multiplication rule: \( P(A\cap B)=P(A)\times P(B|A)=\frac{3}{5}\times\frac{2}{4}=\frac{3}{10} \), and \( P(A)\times P(B)=\frac{3}{5}\times\frac{3}{5}=\frac{9}{25} \). Since \( \frac{3}{10}
eq\frac{9}{25} \), the events are not independent.

Answer:

The events of randomly selecting a white golf ball first and then a white golf ball second are not independent.