# How can you tell the difference between “simple” and “compound” events?

I'll give 5 examples

1. Getting a sum of 10 in a toss of 2 dice

2. Getting a hand of 8 cards consisting of 3 aces and 5 face cards

3. Seating 7 persons in a row, 2 of whom should sit together

4. Getting at least 2 heads in a toss of 4 coins

5. Getting 2 red balls from a box containing 5 red balls and 6 blue balls

I believe 1, 3, and 5 are simple events, while 2 and 4 are compound events

But my problem is that I cannot tell if this is true or not, I'm uncertain of whether an event is simple or compound. How can I tell the difference?

• How do you define "simple event" and "compound event." I personally have never bothered making a distinction as theoretically any event can be expressed as a single set without the need of using multiple sets to do so (even if it may be made more convenient in doing so). – JMoravitz Mar 11 '17 at 8:40
• I believe that's my main problem as well, I can't define the meaning of simple or compound, yet I see it in my text book and no explanation is given. I guess this question is subjective in some way. Sorry about that. – Jan Gamma Mar 11 '17 at 8:42
• Even using a more standard term such as "atomic event" (an event which is a singleton set, containing only a single outcome), this is subjective since we are told nothing of the sample space. A common sample space for rolling two dice is $\{(1,1),(1,2),(1,3),\dots,(2,1),(2,2),\dots,(6,6)\}$ but equally valid sample spaces are $\{2,3,4,\dots,12\}$ and $\{even,odd\}$ as well as $\{10,\text{not}~10\}$ – JMoravitz Mar 11 '17 at 8:45
• At first glance I would distinguish Q4 as needing addition (or subtraction) while I suspect the other questions can be done just by multiplication and division. But then I start wondering about Q1: is it $\frac{2+1}{6^2}$?. – Henry Mar 11 '17 at 10:05
• here's a great resource for intuitively understanding compound probability with the aid of visualizations. students.brown.edu/seeing-theory – Daniel Xiang Mar 11 '17 at 16:20

I looked up the definition of compound event and found my original comment was wrong, but I think I can explain compound event now:

Simply stated, a simple event can happen only one way (has only one simple outcome) whereas a compound event can happen multiple ways (it's a subset of the sample space consisting of more than one simple outcome).

An easy example: The experiment is roll a single die. The sample space (set of all possible simple outcomes) is {1,2,3,4,5,6}. The event '$E$=getting a number more than two and less than four' is a simple outcome, as it can only happen if you get 3 on the die, $E$={3}. However, the event '$F$=getting an even number' is a compound event, because if you get 2, 4 or 6 then the event happens, $F$={2,4,6}.

From this definition, you can show that all of your above example are compound events.

Event 1 = {(4,6),(5,5),(6,4)} three simple outcomes so compound.

Event 2 has $\binom 4 3*\binom {12} 5$ simple outcomes, one of which is aces of diamonds, hearts and clubs, all the jacks, and king of hearts. (I'm assuming this is choosing 8 cards from a standard deck without replacement.)

etc.

Edit: By the way, things do get a bit messy. Let me give an example. Another possible sample space for rolling a die is {even,odd}. It is not useful for finding the probability of rolling a number less than 2, but it works if we want to know the probability of getting an even number. So is the event getting an even number really compound? Shrug.