Simple vs. Compound events and experiments

[jpanknin]

I'm having some trouble understanding the nuances of simple vs. compound events. I've seen the definitions in the form of:

• A simple event can only happen one way such as rolling a die and getting a 3 out of the sample space of S = {1, 2, 3, 4, 5, 6}
• A compound event involves more than one outcome such as rolling an even number E = {2, 4, 6} out of the sample space of S = {1, 2, 3, 4, 5 6}

But what about events such as flipping a coin three times and getting three heads H = {HHH} out of the sample space of 8 possibilities. In some of the websites I've seen this is listed as a "simple event." It seems to me that three "actions" have been taken and that {HHH} is a compound event in the sense that more than one experiment happened (flipping the coin three separate times instead of once).

So why is an event such as {HHH} that requires three experiments considered a simple event? Or would an "experiment" in this case consist of the single result of all three flips? Would {HHH} be considered one experiment or three experiments? If {HHH} is considered a single experiment, what do you call each of the individual coin flips during the experiment?

 

[The Highlander]

I'm having some trouble understanding the nuances of simple vs. compound events. I've seen the definitions in the form of:

• A simple event can only happen one way such as rolling a die and getting a 3 out of the sample space of S = {1, 2, 3, 4, 5, 6}
• A compound event involves more than one outcome such as rolling an even number E = {2, 4, 6} out of the sample space of S = {1, 2, 3, 4, 5 6}

But what about events such as flipping a coin three times and getting three heads H = {HHH} out of the sample space of 8 possibilities. In some of the websites I've seen this is listed as a "simple event." It seems to me that three "actions" have been taken and that {HHH} is a compound event in the sense that more than one experiment happened (flipping the coin three separate times instead of once).

So why is an event such as {HHH} that requires three experiments considered a simple event? Or would an "experiment" in this case consist of the single result of all three flips? Would {HHH} be considered one experiment or three experiments? If {HHH} is considered a single experiment, what do you call each of the individual coin flips during the experiment?

I would just call them "coin flips" and a single event would, indeed, be the result of three coin flips, so {HHH} is just a single experimental result.

Yes, "three "actions" have been taken" but those three actions constitute a single "experiment" or result.

I'm afraid it becomes even trickier if you're asked to evaluate the probability of getting a head & two tails (or vice versa) from three flips of a fair coin! Then your "sample space" is reduced from eight to four (because it no longer matters whether you throw {HTT} or {THT} or {TTH} as all three of those outcomes are the 'same' if you are only considering a head & two tails.

It's the difference between Permutations (where the order matters) and Combinations (where it doesn't).

You might like to have a look at this website for further explanation.

When the order does matter, the possible outcomes are 2³ = 8 (The number of outcomes you mentioned).

But when the order doesn't matter then the number of possible outcomes is calculated as: \(\displaystyle \frac{(3+2-1)!}{3!(2-1)!}\) = 4, ie:-

3 Heads or 3 Tails or 2 Heads & 1 Tail or 2 Tails & 1 Head.​

(NB: 3! = 3 × 2 × 1 = 6 and is called: "3 Factorial" but the website mentioned above explains the Factorial Function if you care to read it; the above formula is also explained there. )

Hope that helps.

 

[Dr.Peterson]

I'm having some trouble understanding the nuances of simple vs. compound events. I've seen the definitions in the form of:

• A simple event can only happen one way such as rolling a die and getting a 3 out of the sample space of S = {1, 2, 3, 4, 5, 6}
• A compound event involves more than one outcome such as rolling an even number E = {2, 4, 6} out of the sample space of S = {1, 2, 3, 4, 5 6}

But what about events such as flipping a coin three times and getting three heads H = {HHH} out of the sample space of 8 possibilities. In some of the websites I've seen this is listed as a "simple event." It seems to me that three "actions" have been taken and that {HHH} is a compound event in the sense that more than one experiment happened (flipping the coin three separate times instead of once).

So why is an event such as {HHH} that requires three experiments considered a simple event? Or would an "experiment" in this case consist of the single result of all three flips? Would {HHH} be considered one experiment or three experiments? If {HHH} is considered a single experiment, what do you call each of the individual coin flips during the experiment?

I suspect the terminology is not entirely consistent.

But to me, a compound event is one that is described using "or" or "and", so that its description has multiple parts. On the other hand, a compound event may result from a simple experiment (rolling one die and getting a 1 or a 3) or a compound experiment (rolling two dice and getting a 1 on the red one and a 3 on the green one). That is, what you do (the one experiment) may consist of multiple actions; and in any case the event may be described in terms of multiple outcomes.

I would probably not describe flipping a coin three times as a simple event; but once you start thinking of a single sample space (as in that "head and two tails" example, where the individual tosses are not distinguished), it might make sense. The distinction is not always meaningful.

 

[Steven G]

I'm having some trouble understanding the nuances of simple vs. compound events. I've seen the definitions in the form of:

• A simple event can only happen one way such as rolling a die and getting a 3 out of the sample space of S = {1, 2, 3, 4, 5, 6}
• A compound event involves more than one outcome such as rolling an even number E = {2, 4, 6} out of the sample space of S = {1, 2, 3, 4, 5 6}

But what about events such as flipping a coin three times and getting three heads H = {HHH} out of the sample space of 8 possibilities. In some of the websites I've seen this is listed as a "simple event." It seems to me that three "actions" have been taken and that {HHH} is a compound event in the sense that more than one experiment happened (flipping the coin three separate times instead of once).

So why is an event such as {HHH} that requires three experiments considered a simple event? Or would an "experiment" in this case consist of the single result of all three flips? Would {HHH} be considered one experiment or three experiments? If {HHH} is considered a single experiment, what do you call each of the individual coin flips during the experiment?

Why are you worrying about this? If you can do the problem correctly is all that matters, at least to me.

 

[jpanknin]

I suspect the terminology is not entirely consistent.

But to me, a compound event is one that is described using "or" or "and", so that its description has multiple parts. On the other hand, a compound event may result from a simple experiment (rolling one die and getting a 1 or a 3) or a compound experiment (rolling two dice and getting a 1 on the red one and a 3 on the green one). That is, what you do (the one experiment) may consist of multiple actions; and in any case the event may be described in terms of multiple outcomes.

I would probably not describe flipping a coin three times as a simple event; but once you start thinking of a single sample space (as in that "head and two tails" example, where the individual tosses are not distinguished), it might make sense. The distinction is not always meaningful.

Yes, I think the terminology is a bit confusing or at least not well-explained in most resources. Thank you for the help.

 

[jpanknin]

Why are you worrying about this? If you can do the problem correctly is all that matters, at least to me.

Because my brain doesn't really understand something until I understand how and why it works rather than just that it works. Even looking up the etymology behind many math terms has been super helpful for me to get a better understanding. Then I find that I don't have to remember formulas because I understand concepts. Or it could also be my OCD that my girlfriend frequently reminds me of...

 

[Steven G]

Because my brain doesn't really understand something until I understand how and why it works rather than just that it works. Even looking up the etymology behind many math terms has been super helpful for me to get a better understanding. Then I find that I don't have to remember formulas because I understand concepts. Or it could also be my OCD that my girlfriend frequently reminds me of...

Yes, you should understand the material and know why it works. I 100% agree with that. If you understand the pythagorean theorem, for example, I do not understand why knowing the name of this theorem is helpful (other than with communicating with others).

 

[jpanknin]

Yes, you should understand the material and know why it works. I 100% agree with that. If you understand the pythagorean theorem, for example, I do not understand why knowing the name of this theorem is helpful (other than with communicating with others).

I have always tried to take examples from people much smarter than I am hoping that some of it might transfer over. In this case, I think the best explanation I can give is quoting Richard Feynman from the video, "The Pleasure of Finding Things Out." If you haven't watched it, it's pretty amazing and freely available. He says:

"I have a friend who’s an artist and he’s sometimes taken a view which I don’t agree with very well. He’ll hold up a flower and say, “Look how beautiful it is,” and I’ll agree, I think. And he says-“you see, I as an artist can see how beautiful this is, but you as a scientist, oh, take this all apart and it becomes a dull thing.” And I think that he’s kind of nutty. First of all, the beauty that he sees is available to other people and to me, too, I believe, although I might not be quite as refined aesthetically as he is; but I can appreciate the beauty of a flower. At the same time I see much more about the flower than he sees. I can imagine the cells in there, the complicated actions inside which also have a beauty. I mean it’s not just beauty at this dimension of one centimeter, there is also beauty at a smaller dimension, the inner structure. Also the processes, the fact that the colors in the flower evolved in order to attract insects to pollinate it is interesting-it means that insects can see the color. It adds a question: Does this aesthetic sense also exist in the lower forms?Why is it aesthetic? All kinds of interesting questions which shows that a science knowledge only adds to the excitement and mystery and the awe of a flower. It only adds; I don’t understand how it subtracts."

I find that beauty in understanding something so completely that I don't have to even think about it anymore. That's probably why I'm awake learning about Abelian groups, rings, and fields at 1am on a Friday night. So perhaps whether this is a good thing is subjective.

 

[Steven G]

I am not disagreeing with you about the beauty of math and that one should understand math. Does it really make a difference that they call a commutative group an abelian group? What if they decide to call an a commutative group a dabba group. Would anything at all change? All I am saying is that in my opinion, what they call something doesn't matter at all!
Think about this. Suppose you study math in German and I study math in English. Must you, who studies in German, be better at math than I am. After all, your words are different. In particular, their math words are different. You seem to be saying that what we call things matter. That's not true at all. Concepts matter, not the name of something.

 

[jpanknin]

I am not disagreeing with you about the beauty of math and that one should understand math. Does it really make a difference that they call a commutative group an abelian group? What if they decide to call an a commutative group a dabba group. Would anything at all change? All I am saying is that in my opinion, what they call something doesn't matter at all!
Think about this. Suppose you study math in German and I study math in English. Must you, who studies in German, be better at math than I am. After all, your words are different. In particular, their math words are different. You seem to be saying that what we call things matter. That's not true at all. Concepts matter, not the name of something.

It depends. Sometimes I've found the etymology of words helpful, like tangent. Others like sine comes from sinus, which apparently in Latin means "fold in a garment, bend, curve, bosom" provided no help at all and actually made things more confusing. Or Abelian because it was named after Niels Abel also provides no help. But sometimes it does.