# Introduction to probability theory: word expression for A - BC

A simple question: if $$A$$ is the event of a man who is older than 40, $$B$$ is the event where he is older than his wife, and $$C$$ is the event where his wife is older than 40. How can the event $$A - BC$$ be described in word? My attempt was: the husband is older than 40 but either his wife maximum age is 40 or she is older than him. Was it correct? The reason I ask this question is I want to be able to translate a probability question back to a mathematical expression. Thanks.

EDIT

Visually, $$A$$ is the area on the right side of the red broken line, $$BC$$ is the triangular area formed by black and blue broken line. Thus, $$A - BC$$ would include any O points but not X point. As the result, I described $$A - BC$$ as "the husband was older than 40, whilst either his wife maximum age was 40 or she was older than him."

Very sorry for the confusion, I had a mistake in my former version of this. Here is my revision.

Event $$BC$$ contradicts $$A,$$ because $$BC$$ implies that the man is over 40. Thus $$A - BC$$ is null. More formally:

$$D = BC = \{M>W, W>40\} \subset \{M > 40\} = A$$

because the former condition implies the latter. Then $$A - BC = A - D = AD^c = \emptyset.$$

• Hi Bruce, as my OP was: how to describe in (English) words the mathematical expression $A - BC$; whilst your answer showed $A - BC = A$. Did you mean the description I was looking for was: "the event of husband is older than 40"? If so, did your answer imply that my interpretation of "the husband is older than 40 but either his wife younger than 40 or she is older than him" was not correct? Thanks.
– Nemo
Commented Jan 4, 2021 at 3:08
• Thanks for the clarification, Bruce. Taking time to digest your somersaulting expression above, I think I can now understand the answer more. How could I learn such method to transform from $A - BC$ to $A \cap \Omega$ like you did?
– Nemo
Commented Jan 4, 2021 at 11:27
• Very sorry. My initial try at the was wrong. Reviewing my notes this morning, I see that I had a mistake among the M's and W's that led to an incorrect last line of what I posted before. Feel free to retract you acceptance and upvote, if you don't feel the revision answers your question. // In plain English the statement is contradictory because it says that the man is and is not over 40. Commented Jan 4, 2021 at 19:24
• Thanks for the rectification. What do you think of my reasoning in the edited post, Bruce?
– Nemo
Commented Jan 5, 2021 at 4:10