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 is found in the word " \( B E A D^{\text {" }} \). Event \( Y: \) The letter selected comes after " \( D \) ". 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 \( Y \) ": \{[i]) (b) Event \( { }^{n} X \) or \( Y *:\{D\} \) (c) The complement of the event \( Y:\{\square\} \)

To solve the problem, we first identify the letters on the tiles and the events defined. The letters on the tiles are: \( A, B, C, D, E, F, G \). ### Event Definitions: - **Event \( X \)**: The letter selected is found in the word "BEAD". The letters in "BEAD" are \( B, E, A, D \). - **Event \( Y \)**: The letter selected comes after "D". The letters that come after "D" are \( E, F, G \). Now, let's find the outcomes for each of the specified events. ### (a) Event " \( X \) and \( Y \) " This event represents the intersection of events \( X \) and \( Y \). We need to find the letters that are both in \( X \) and \( Y \). - Outcomes of \( X \): \( \{B, E, A, D\} \) - Outcomes of \( Y \): \( \{E, F, G\} \) The intersection \( X \cap Y \) is: - Common letters: \( E \) Thus, the outcome for event " \( X \) and \( Y \) " is: \[ \{E\} \] ### (b) Event " \( X \) or \( Y \) " This event represents the union of events \( X \) and \( Y \). We need to find all unique letters that are in either \( X \) or \( Y \). - Outcomes of \( X \): \( \{B, E, A, D\} \) - Outcomes of \( Y \): \( \{E, F, G\} \) The union \( X \cup Y \) is: - Unique letters: \( B, E, A, D, F, G \) Thus, the outcome for event " \( X \) or \( Y \) " is: \[ \{B, E, A, D, F, G\} \] ### (c) The complement of the event \( Y \) The complement of event \( Y \) includes all letters that are not in \( Y \). - Outcomes of \( Y \): \( \{E, F, G\} \) - The complement of \( Y \) will include the remaining letters: \( A, B, C, D \) Thus, the outcome for the complement of event \( Y \) is: \[ \{A, B, C, D\} \] ### Summary of Outcomes: - (a) Event " \( X \) and \( Y \) ": \(\{E\}\) - (b) Event " \( X \) or \( Y \) ": \(\{B, E, A, D, F, G\}\) - (c) The complement of the event \( Y \): \(\{A, B, C, D\}\)