Certainly! Let's go through the solution step-by-step.
(a) To find the probabilities of each event:
- Event A: Drawing a grey ball.
- Event B: Drawing a white ball.
Total number of outcomes: 8
Grey balls: Numbered [tex]\(1, 2, 5, 3\)[/tex]. So, there are 4 grey balls.
White balls: Numbered [tex]\(3, 4, 6, 8\)[/tex]. So, there are 4 white balls.
i. Probability of Event [tex]\(A\)[/tex] (drawing a grey ball):
[tex]\[ P(A) = \frac{\text{Number of grey balls}}{\text{Total number of outcomes}} = \frac{4}{8} = 0.5 \][/tex]
ii. Probability of Event [tex]\(B\)[/tex] (drawing a white ball):
[tex]\[ P(B) = \frac{\text{Number of white balls}}{\text{Total number of outcomes}} = \frac{4}{8} = 0.5 \][/tex]
iii. Probability of Event [tex]\(A \text{ and } B\)[/tex] (a ball that is both grey and white):
Notice that ball number 3 is both grey and white, so:
[tex]\[ P(A \text{ and } B) = \frac{\text{Number of balls that are both grey and white}}{\text{Total number of outcomes}} = \frac{1}{8} = 0.125 \][/tex]
(b) Compute the combined probability:
Using the formula for the union of two events:
[tex]\[ P(A \cup B) = P(A) + P(B) - P(A \text{ and } B) \][/tex]
Plugging in the values:
[tex]\[ P(A \cup B) = 0.5 + 0.5 - 0.125 = 0.875 \][/tex]
So, the combined probability is [tex]\(0.875\)[/tex].
(c) Select the answer that makes the equation true:
Given:
[tex]\[ P(A) + P(B) - P(A \text{ and } B) = 0.875 \][/tex]
The answer that makes the equation true is [tex]\(0.875\)[/tex].
In summary:
1. [tex]\( P(A) = 0.5 \)[/tex]
2. [tex]\( P(B) = 0.5 \)[/tex]
3. [tex]\( P(A \text{ and } B) = 0.125 \)[/tex]
4. [tex]\( P(A \cup B) = 0.875 \)[/tex]
So, the completed answers are:
[tex]\[
\begin{aligned}
&\text{(a) } P(A) = 0.5, P(B) = 0.5, P(A \text{ and } B) = 0.125 \\
&\text{(b) } P(A) + P(B) - P(A \text{ and } B) = 0.875 \\
&\text{(c) } \text{The answer that makes the equation true is } 0.875
\end{aligned}
\][/tex]
