# Question about sample spaces and events

• Jul 11th 2011, 10:09 PM
downthesun01
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.
• Jul 12th 2011, 01:37 AM
Plato
Re: Question about sample spaces and events
Quote:

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)\$
• Jul 12th 2011, 01:53 AM
downthesun01
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.
• Jul 12th 2011, 02:01 AM
Plato
Re: Question about sample spaces and events
Quote:

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.
• Jul 12th 2011, 02:28 AM
downthesun01
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\$?
• Jul 12th 2011, 03:00 AM
Plato
Re: Question about sample spaces and events
Quote:

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?
• Jul 12th 2011, 08:40 PM
downthesun01
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