To solve the problem of finding the probability of either event \(A\) or event \(B\) or both occurring, we can use the formula for the union of two events. The formula is given as follows:
[tex]\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \][/tex]
Where:
- \( P(A) \) is the probability of event \(A\) occurring.
- \( P(B) \) is the probability of event \(B\) occurring.
- \( P(A \cap B) \) is the probability of both events \(A\) and \(B\) occurring simultaneously.
Given:
[tex]\[ P(A) = \frac{1}{4}, \][/tex]
[tex]\[ P(B) = \frac{1}{6}, \][/tex]
[tex]\[ P(A \cap B) = \frac{1}{24} \][/tex]
First, convert the fractions to decimal form for clarity:
[tex]\[ P(A) = \frac{1}{4} = 0.25 \][/tex]
[tex]\[ P(B) = \frac{1}{6} \approx 0.16666666666666666 \][/tex]
[tex]\[ P(A \cap B) = \frac{1}{24} \approx 0.041666666666666664 \][/tex]
Now, substitute these values into the formula:
[tex]\[ P(A \cup B) = 0.25 + 0.16666666666666666 - 0.041666666666666664 \][/tex]
Let's perform the addition and subtraction step by step:
1. Add \( P(A) \) and \( P(B) \):
[tex]\[ 0.25 + 0.16666666666666666 = 0.41666666666666666 \][/tex]
2. Subtract \( P(A \cap B) \) from the result:
[tex]\[ 0.41666666666666666 - 0.041666666666666664 \approx 0.37499999999999994 \][/tex]
Therefore, the probability of either event \(A\) or event \(B\) or both occurring is:
[tex]\[ P(A \cup B) \approx 0.375 \][/tex]
In fraction form, this probability is:
[tex]\[ P(A \cup B) = \frac{3}{8} = 0.375 \][/tex]
So, the probability of [tex]\(A\)[/tex] or [tex]\(B\)[/tex] or both is approximately [tex]\(0.375\)[/tex] or [tex]\(37.5\%\)[/tex].
