In Problems 5–12, 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}S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}S={1,2,3,4,5,6,7,8,9,10,11,12}. Let event E={2,3,4,5,6,7}E = \{2, 3, 4, 5, 6, 7\}E={2,3,4,5,6,7}, event F={5,6,7,8,9}F = \{5, 6, 7, 8, 9\}F={5,6,7,8,9}, event G={9,10,11,12}G = \{9, 10, 11, 12\}G={9,10,11,12}, and event H={2,3,4}H = \{2, 3, 4\}H={2,3,4}. Assume that each outcome is equally likely.
List the outcomes in Fc. Find P(Fc).

Verified step by step guidance

1. Identify the sample space \(S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}\) and the event \(F = \{5, 6, 7, 8, 9\}\).
2. Recall that the complement of event \(F\), denoted \(F^c\), consists of all outcomes in the sample space \(S\) that are NOT in \(F\).
3. List the outcomes in \(F^c\) by selecting all elements from \(S\) that are not in \(F\). This means removing \$5, 6, 7, 8,\( and \)9\( from \)S$.
4. Count the number of outcomes in \(F^c\) to find \(n(F^c)\), and also note the total number of outcomes in the sample space \(n(S)\).
5. Calculate the probability of the complement event \(F^c\) using the formula \(P(F^c) = \frac{n(F^c)}{n(S)}\), since all outcomes are equally likely.

Key Concepts

Sample Space and Events

The sample space is the set of all possible outcomes of a probability experiment. An event is a subset of the sample space, representing outcomes of interest. Understanding the sample space and events helps identify which outcomes belong to each event and is essential for calculating probabilities.

Complement of an Event

The complement of an event F, denoted Fc, consists of all outcomes in the sample space that are not in F. It represents the event 'F does not occur.' Knowing how to find the complement is crucial for solving problems involving probabilities of events not happening.

Probability of an Event with Equally Likely Outcomes

When all outcomes in the sample space are equally likely, the probability of an event is the ratio of the number of outcomes in the event to the total number of outcomes in the sample space. This principle allows straightforward calculation of probabilities by counting outcomes.