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 Ec. Find P(Ec).

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 \(E = \{2, 3, 4, 5, 6, 7\}\).
2. Recall that the complement of event \(E\), denoted \(E^c\), consists of all outcomes in the sample space \(S\) that are not in \(E\).
3. List the outcomes in \(E^c\) by selecting all elements from \(S\) that are not in \(E\). This means \(E^c = S \setminus E\).
4. Count the number of outcomes in \(E^c\) to find \(n(E^c)\) and the total number of outcomes in \(S\) to find \(n(S)\).
5. Calculate the probability of the complement event \(E^c\) using the formula \(P(E^c) = \frac{n(E^c)}{n(S)}\), 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 a particular event or its complement.

Complement of an Event

The complement of an event E, denoted Ec, consists of all outcomes in the sample space that are not in E. It represents the event 'E does not occur.' Finding Ec involves subtracting the outcomes of E from the sample space.

Calculating Probability 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 favorable outcomes to the total number of outcomes. This principle allows straightforward calculation of probabilities by counting outcomes.