# How is a set an event

For problems where sets are described as events such as:

In a group of 12 friends, 8 study French, 4 study french only and 2 study neither French nor Cantonese. Let A be the event 'studies French' and let B be the event 'studies Cantonese'.

To my mind, A and B are sets and the elements are people (friends). How are they then also events? Is it because the 12 friends comprise the "sample space" and the event is "sampling"?

A set is any collection of objects (mathematical or not). An event is "an outcome or defined collection of outcomes of a random experiment" (well, according to Statistics.com). I don't think it's a contentious subject so I'll quote wikipedia here for the definition of an event: "In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned."

Where is the experiment in the initial statements? Or am I confusing the sets defined by the text with my own e.g.

Set A={a,b,c,d,e,f,g,h} Set B={e,f,g,h,i,j} compliment of the sets = {k,l}

Perhaps the categories are the 'elements' of the sets in this instance? Set A={any friend who studies French} Set B={any friend who studies Cantonese} compliment of the sets = {any friend who studies neither}

I am new to this but I've heard of set builder notation. I'm wondering how to describe "Let A be the event 'studies French' and let B be the event 'studies Cantonese'." in set builder in that case.

• Events are sets in probability that contain outcomes. That's just the language we use. Jul 17, 2022 at 2:19
• Yes… I see. I think the text I am working out of, usually it is very good, but in this case they are using contracted expressions, wherein “Let event A=studies Mandarin” should be written “let Event A=a randomly selected friend studies Mandarin” Jul 17, 2022 at 4:25
• I’ve come to the conclusion that this passage in the text may simply be poorly written. Jul 17, 2022 at 20:43

In a group of 12 friends, 8 study French, 4 study French only and 2 study neither French nor Cantonese. Let A be the event 'studies French' and let B be the event 'studies Cantonese'.

To my mind, A and B are sets and the elements are people (friends). How are they then also events? Is it because the event is "sampling"?

I'm wondering how to describe "Let A be the event 'studies French' in set builder notation

How an event is defined depends on how the experiment is set up. (Click here for my detailed explanation.)

It appears that in this experiment, $$A$$ is the event that the chosen friend studies French, and equals $$\{x\mid x\text{ studies French}\}$$ (equivalently, list those four friends within the braces).

In a different experiment, {Jill-Bill, Jill-Phil} might be the event of both chosen friends studying French. Notice that each outcome in this experiment comprises two names; this is because, unlike the above single-trial experiment, this experiment has two trials.

Set A={any friend who studies French}

If you'd like to describe the set (in lieu of either option above), replace “any friend who” with “all friends who”, and omit that pair of braces.

"An event is an outcome or defined collection of outcomes of a random experiment" (well, according to Statistics.com).

We usually abuse notation, writing $$P(HTT)$$ and $$P(HTT,THT,TTH)$$ instead of $$P(\{HTT\})$$ and $$P(\{HTT,THT,TTH\}).$$ But, technically, an elementary event (an event with a single element) is still a set, rather than an outcome.

I'll quote wikipedia: "an event is a a subset of the sample space"

Yes, exactly. And the sample space varies with the experiment.