A probability experiment is conducted in which the sample space of the experiment is \( S=\{1,2,3,4,5,6,7,8,9,10,11,12\} \), event \( F=\{5,6,7,8,9,10\} \), and event \( G=\{9,10,11,12\} \). Assume that each outcome is equally likely. List the outcomes in \( F \) or \( G \). Find \( P(F \) or \( G) \) by counting the number of outcomes in \( F \) or \( G \). Determine \( P(F \) or \( G) \) using the general addition rule. List the outcomes in \( F \) or \( G \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. F or \( G=\{5,6,7,8,9,10,11,12\} \) (Use a comma to separate answers as needed.) Find \( P(F \) or \( G=\{ \} \) \( P(F \) or \( G) \) by counting the number of outcomes in \( F \) F or \( G \). (Type an integer or a decimal rounded to three decimal places as needed.) Determine \( P(F \) or \( G) \) using the general addition rule. Select the correct choice below and fill in any answer boxes within your choice. (Type the terms of your expression in the same order as they appear in the original expression. Round to three decimal places as needed.) A. \( P(F \) or \( G)=\square+\square-\square=\square \) B. \( P(F \) or \( G)=\square+\square=\square \)
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First, the outcomes in \( F \) or \( G \) would be the union of the two sets, which combines all unique elements from both \( F \) and \( G \): \( F \) or \( G = \{5, 6, 7, 8, 9, 10, 11, 12\} \). Next, to find \( P(F \) or \( G) \), we first count the outcomes. The total number of outcomes is 12 (from \( S \)), and the number of outcomes in \( F \) or \( G \) is 8. Therefore, \( P(F \) or \( G) = \frac{8}{12} = \frac{2}{3} \approx 0.667 \). Now, using the general addition rule, we calculate: \( P(F) = \frac{6}{12} = 0.500 \), \( P(G) = \frac{4}{12} = 0.333 \), and \( P(F \cap G) = P(\{9, 10\}) = \frac{2}{12} = \frac{1}{6} \approx 0.167 \). Using these, we have: A. \( P(F \) or \( G) = P(F) + P(G) - P(F \cap G) = \frac{6}{12} + \frac{4}{12} - \frac{2}{12} = \frac{8}{12} = \frac{2}{3} \). So rounding to three decimal places, A. \( P(F \) or \( G)=0.500+0.333-0.167=0.667 \).
