1. ## Question about sample spaces and events

Let E, F, G be three events. Find expressions for the events that of E, F, G
a. only F occurs
b. Both E and F but not G occur
c. at least one event occurs
d. at least two events occur
e. all three events occur
f. none occurs
g. at most one occurs
h. at most two occur

The word "expressions" is confusing me a little bit here.

This is what I have right now, but I'm not sure how right it all is:
a. $\displaystyle F\cap(G\cup E)^c$
b. $\displaystyle (E\cap F)\cup G^c)$
c. This is one I'm not sure about, simply because of how long it is
$\displaystyle [(F\cap(G\cup E)^c)\cup (G\cap(F\cup E)^c)\cup (E\cap(G\cup F)^c)\cup((E\cap F)\cup G^c))\cup((E\cap G)\cup F^c))\cup((G\cap F)\cup E^c)) \cup(E\cap F\cap G)]\cap (E \cup F\cup G)^c)$
d.

I don't even want to bother putting the rest in because it just seems like I'm way wrong.

2. ## Re: Question about sample spaces and events

Originally Posted by downthesun01
Let E, F, G be three events. Find expressions for the events that of E, F, G
a. only F occurs
b. Both E and F but not G occur
c. at least one event occurs
d. at least two events occur
e. all three events occur
f. none occurs
g. at most one occurs
h. at most two occur
You are really over thinking most of these.
I will do three of these, you show the reat,

b) $\displaystyle E\cap F\cap G^c$

c) $\displaystyle E\cup F\cup G$

d) $\displaystyle (E\cap F)\cup(E\cap G)\cup(F\cap G)$

3. ## Re: Question about sample spaces and events

You're right, I was over-thinking this all.

So,

e. is $\displaystyle E\cap F\cap G$
f. is $\displaystyle (E\cap F\cap G)^c$
g. is $\displaystyle ((E\cap F\cap G)^c)\cup E\cup F\cup G$
h. is $\displaystyle (E\cap F)\cup (E\cap G)\cup (F\cap G)\cup (E\cap F\cap G)^c)$

I hope that's all right.

Looking at it, I'm pretty sure g and h are wrong still.

4. ## Re: Question about sample spaces and events

Originally Posted by downthesun01
e. is $\displaystyle E\cap F\cap G$
f. is $\displaystyle \color{red}(E\cap F\cap G)^c$
Do you see that f) is the complement of c)?
What you have above is "At least one does not occur."

The others need work as well.

5. ## Re: Question about sample spaces and events

I'm sorry. Is E correct?

I see what you mean about F. it should be $\displaystyle (E\cup F\cup G)^c$, right?

G. is $\displaystyle (E\cup F\cup G)^c\cup E\cup F\cup G$?

Wow, I'm confusing myself so much right now.

Is H. $\displaystyle E^c \cup F^c \cup G^c$?

And G. $\displaystyle (E\cap F)^c \cup (E\cap G)^c \cup (F\cap G)^c$?

6. ## Re: Question about sample spaces and events

Originally Posted by downthesun01
Is E correct?
Yes, E) is correct.
These pairs are complements of each other: $\displaystyle f\& c,~g\& d,~h\& e$.

Can you see that?

7. ## Re: Question about sample spaces and events

Sorry it took so long to reply.

Yes, after getting some sleep, I can see that what you mean. Thank you for your help