work \( 10(5.1,5.2) \) Question 33, 5.2.7 HW Score: \( 66 \%, 33 \) of 50 points Part 3 of 3 Points: 0 of 1 Save A probability experiment is conducted in which the sample space of the experiment is \( S=\{9,10,11,12,13,14,15,16,17,18,19,20\} \), event \( F=\{11,12,13,14,15\} \), and event \( G=\{15,16,17,18\} \). 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 \( G=\{11,12,13,14,15,16,17,18\} \) (Use a comma to separate answers as needed.) B. \( F \) or \( G=\{ \} \) Find \( \mathrm{P}(\mathrm{F} \) or G\( ) \) by counting the number of outcomes in F or G . \[ P(F \text { or } G)=0.667 \] (Type an integer or a decimal rounded to three decimal places as needed.) Determine \( \mathrm{P}(\mathrm{F} \) 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. \( \mathrm{P}(\mathrm{F} \) or G\( )=\square+\square-\square=\square \) \( \square \) \( \square \) \( \square \) B. \( P(F \) or \( G)= \) \( \square \) \( + \) \( = \) \( \square \)
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To find the outcomes in \( F \) or \( G \), we combine the elements of both sets while ensuring we don't double-count any shared elements. The union of sets \( F \) and \( G \) gives: \[ F \text{ or } G = \{11, 12, 13, 14, 15\} \cup \{15, 16, 17, 18\} = \{11, 12, 13, 14, 15, 16, 17, 18\} \] Now, for the probability \( P(F \text{ or } G) \), we need to count the outcomes. There are 8 outcomes in \( F \text{ or } G \) and a total of 12 outcomes in the sample space \( S \): \[ P(F \text{ or } G) = \frac{8}{12} = \frac{2}{3} = 0.667 \] Using the general addition rule to determine \( P(F \text{ or } G) \): \[ P(F \text{ or } G) = P(F) + P(G) - P(F \text{ and } G) \] Calculating each term: - \( P(F) = \frac{5}{12} \) - \( P(G) = \frac{4}{12} \) - \( P(F \text{ and } G) = \frac{1}{12} \) because the only common element is 15. Thus, we have: \[ \begin{align*} P(F \text{ or } G) &= \frac{5}{12} + \frac{4}{12} - \frac{1}{12} \\ &= \frac{8}{12} \\ &= 0.667 \end{align*} \] So the correct answers are: **A**. \( F \text{ or } G=\{11,12,13,14,15,16,17,18\} \) **B**. \( P(F \text{ or } G) = \frac{5}{12} + \frac{4}{12} - \frac{1}{12} = \frac{8}{12} \) (or \( 0.667 \))
