A probability experiment is conducted in which the sample space of the experiment is [tex]$S = \{5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}$[/tex], event [tex]$F = \{9, 10, 11, 12, 13, 14\}$[/tex], and event [tex]$G = \{13, 14, 15, 16\}$[/tex]. Assume that each outcome is equally likely. List the outcomes in F or G. Find [tex]$P(F \text{ or } G)$[/tex] by counting the number of outcomes in F or G. Determine [tex]$P(F \text{ or } G)$[/tex] 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 [tex]$G = \boxed{\ }[/tex]. B. F or [tex]$G = \{\ \}[/tex] Find [tex]$P(F \text{ or } G)$[/tex] by counting the number of outcomes in F or G. [tex]$P(F \text{ or } G) = \square[/tex] (Type an integer or a decimal rounded to three decimal places as needed.) Determine [tex]$P(F \text{ or } G)$[/tex] using the general addition rule. Select the correct choice below and fill in any answer boxes within your choice. (Type the terms of your expression in the same order as they appear in the original expression. Round to three decimal places as needed.) A. [tex]$P(F \text{ or } G) = \square + \square = \square[/tex] B. [tex]$P(F \text{ or } G) = \square + \square - \square = \square[/tex]
Answer (2)
The outcomes in F or G are {9, 10, 11, 12, 13, 14, 15, 16}, leading to a probability of approximately 0.667. This was found both by counting outcomes and using the general addition rule. Thus, the answer for F or G is A: { 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 } .
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List the outcomes in F or G: F or G = { 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 } .
Find P ( F or G ) by counting: P ( F or G ) = 12 8 = 0.667 .
Determine P ( F or G ) using the general addition rule: P ( F or G ) = P ( F ) + P ( G ) − P ( F and G ) = 12 6 + 12 4 − 12 2 .
Simplify the expression: P ( F or G ) = 12 8 = 0.667 .
Explanation
Understand the problem We are given the sample space S = { 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 } , event F = { 9 , 10 , 11 , 12 , 13 , 14 } , and event G = { 13 , 14 , 15 , 16 } . Our goal is to find the outcomes in F or G , the probability P ( F or G ) by counting outcomes, and the probability P ( F or G ) using the general addition rule.
List outcomes in F or G The outcomes in F or G are the elements that are in either F or G or both. Combining the elements of F and G , we have F or G = { 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 } .
Find P(F or G) by counting To find P ( F or G ) by counting outcomes, we first find the number of outcomes in F or G , which is n ( F or G ) = 8 . The total number of outcomes in the sample space S is n ( S ) = 12 . Therefore, P ( F or G ) = n ( S ) n ( F or G ) = 12 8 = 3 2 ≈ 0.667 .
Determine P(F or G) using the general addition rule Using the general addition rule, we have P ( F or G ) = P ( F ) + P ( G ) − P ( F and G ) . First, we find P ( F ) = n ( S ) n ( F ) = 12 6 = 2 1 . Next, we find P ( G ) = n ( S ) n ( G ) = 12 4 = 3 1 . The intersection of F and G is F and G = { 13 , 14 } , so P ( F and G ) = n ( S ) n ( F and G ) = 12 2 = 6 1 . Therefore, P ( F or G ) = 12 6 + 12 4 − 12 2 = 12 8 = 3 2 ≈ 0.667 .
Final Answer The outcomes in F or G are { 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 } . The probability P ( F or G ) by counting outcomes is approximately 0.667 . Using the general addition rule, P ( F or G ) = P ( F ) + P ( G ) − P ( F and G ) = 12 6 + 12 4 − 12 2 = 12 8 ≈ 0.667 .
Examples
Understanding probability is crucial in many real-life scenarios. For instance, when playing a game of cards, knowing the probability of drawing a specific card can inform your strategy. Similarly, in weather forecasting, probabilities are used to predict the likelihood of rain or sunshine, helping people plan their day. In finance, probabilities are used to assess the risk associated with investments, guiding decisions on where to allocate resources. These examples highlight how probability calculations, like the one we just did, are fundamental in making informed decisions in various aspects of life.
