Question

a tile is selected from seven tiles, each labeled with a different letter from the first seven letters of the alphabet. the letter selected will be recorded as the outcome. consider the following events. event ( x ): the letter selected comes before \( d )\. event ( y ): the letter selected is found in the word \( cage )\. give the outcomes for each of the following events. if there is more than one element in the set, separate them with commas. (a) event \( x ) and ( y )\: (b) event \( x ) or ( y )\: (c) the complement of the event ( y ):

Explanation:

Step1: Determine the sample space

The first seven letters of the alphabet are \(A, B, C, D, E, F, G\). So the sample space \(S=\{A,B,C,D,E,F,G\}\)

Step2: Find the elements of event \(X\)

Event \(X\): The letter selected comes before \(D\). So \(X = \{A,B,C\}\)

Step3: Find the elements of event \(Y\)

Event \(Y\): The letter selected is found in the word “\(CAGE\)”. So \(Y=\{C,A,G,E\}\)

Part (a)

Step1: Find \(X\cap Y\) (Event “\(X\) and \(Y\)”)

\(X\cap Y\) is the set of elements that are in both \(X\) and \(Y\).
\(X\cap Y=\{A,C\}\)

Part (b)

Step1: Find \(X\cup Y\) (Event “\(X\) or \(Y\)”)

\(X\cup Y\) is the set of elements that are in \(X\) or \(Y\) (or both).
\(X\cup Y = \{A,B,C,E,G\}\)

Part (c)

Step1: Find \(\overline{Y}\) (Complement of event \(Y\))

The complement of \(Y\), \(\overline{Y}=S - Y\)
\(\overline{Y}=\{B,D,F\}\)

Answer:

(a) \(\{A,C\}\)
(b) \(\{A,B,C,E,G\}\)
(c) \(\{B,D,F\}\)