The problem involves selecting a tile from seven tiles, each labeled with a different letter from the first seven letters of the alphabet: A, B, C, D, E, F, G.

Definitions:

• Event X: The letter selected comes before "D" (i.e., A, B, C).
• Event Y: The letter selected is found in the word "CAGE" (i.e., C, A, G, E).

We need to determine the outcomes for the following:

(a) Event "X and Y"

The intersection of X and Y is the set of letters that are both before "D" and in "CAGE". This means:

• X (letters before "D") = {A, B, C}
• Y (letters in "CAGE") = {C, A, G, E}

The common letters in both sets are A and C. Therefore:

• Event "X and Y" = {A, C}.

(b) Event "X or Y"

This is the union of X and Y, which means all letters that are either in X or in Y. The union of the two sets is:

• X (letters before "D") = {A, B, C}
• Y (letters in "CAGE") = {C, A, G, E}

The union is {A, B, C, G, E}. Therefore:

• Event "X or Y" = {A, B, C, G, E}.

(c) The complement of Event Y

The complement of Y includes all letters that are not in Y (i.e., not in "CAGE"). The full set of letters is {A, B, C, D, E, F, G}, and Y = {C, A, G, E}. Therefore, the complement of Y includes the letters that are not in Y:

• The complement of Y = {B, D, F}.

Final answers:

• (a) Event "X and Y" = {A, C}
• (b) Event "X or Y" = {A, B, C, G, E}
• (c) Complement of event Y = {B, D, F}

Would you like any further details or clarifications?

Related Questions:

1. What is the difference between union and intersection in probability?
2. How do we calculate the complement of an event?
3. How do we define mutually exclusive events?
4. How would the outcome change if we had 10 letters instead of 7?
5. How is the probability calculated if all letters are equally likely?

Tip:

When solving probability problems, organizing outcomes into sets and using Venn diagrams can help visualize intersections and unions easily.