# Probability question - confusion on what it is actually asking and the distinction between two parts of the question

A coin is tossed three times. Define an appropriate sample space for the following cases:

1) The outcome of each individual toss is of interest

2) Only the number of trials is of interest

I'm a little confused about what it is actually asking for. I presume that the entire sample space is something like this:

$$\{HHH, TTT, HTT, HHT, HTH, TTH, THT, THH\}$$

• Seems more likely second one asks for number of heads. Clearly number of trials is 3... – coffeemath Sep 14 '19 at 1:43
• While not an exact duplicate, I strongly recommend reading my answer here where I discuss how in an experiment we have a choice of what sample space to actually use and how some choices are often seen as more convenient than others. – JMoravitz Sep 14 '19 at 1:44

I strongly recommend you read my answer to the question Why is flipping a head then a tail a different outcome than flipping a tail then a head?

Reiterating a few of the key points, when performing an experiment we record the results of the experiment and these different results are the "outcomes" of the experiment. Specifically, a sample space is the set of all possible outcomes satisfying some rules. In particular, any time we perform the experiment we have an unambiguous single outcome to describe the results.

Something that many people fail to realize however is that we have a choice as to what information is relevant to keep when describing an outcome and as such we have a choice as to what sample space to use for an experiment. When playing a game of monopoly for instance, we could keep track of every single dice roll, every single game action, every single time someone had to pay tax and so on and we can ask questions like "What is the probability that a player lands on park place on the twenty-seventh turn (assuming perfect play)?" On the other hand, if all you are interested in is what the probability that a specific player wins the game and all other intermediate information is irrelevant to you, then the outcomes may simply only reflect that information alone.

The only other real rules to follow for your sample space is that any probability question you wish to ask about your experiment must be uniquely and unambiguously described as a subset of the sample space. So, in our monopoly example, if we had as our sample space only $$\{\text{player A wins, player B wins, player C wins}\}$$ then this is not a sufficient sample space to be used to answer the question of "What is the probability that the player who rolled the most sixes wins?"

There are suggestions however that you should often follow. If you use a finite sample space whose outcomes are all equally likely to occur, then you can take advantage of this by noting that the probability of an event occurring is simply the ratio of the size of the event to the size of the sample space, i.e. $$Pr(A)=\dfrac{|A|}{|S|}$$. Note that this is not true of sample spaces which are not equiprobable, for example when playing the lottery you either win or you lose but winning does not occur with probability $$\frac{1}{2}$$.

The other suggestion is that you can use a sample space which is made simpler as much as possible, leaving out unnecessary information. For instance, when shuffling a deck of cards, you might only be interested in the second card alone without caring about any of the others. The sample space could describe the result of the second card only and that would be sufficient for your needs. There is no need to describe the position of every card in the deck.

These two suggestions can go against one another, so pick the one that is more useful to you at the time.

As to your specific problem, if the outcome of every individual toss is relevant, then the only reasonable sample space is as you say $$\{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$$ where each outcome is a string of characters representing the face of the coin which occurred in each respective point in time.

For the second part I assume first that there is a typo or a missing clarification. I will reword it to what I think it should be and that is "Only the number of trials which resulted in heads is of interest", in which case using that same sample space is certainly a reasonable (and perhaps even preferred) choice. It has the advantage of being an equiprobable sample space which is a particularly useful thing as it allows us to use counting techniques which would otherwise not be available. However, another reasonable choice and probably the one they are implying they are looking for would be the sample space which simply just counts the number of heads and records no additional information. Here then, we could have used the sample space $$\{0,1,2,3\}$$. This has the advantage of being easier to read, and many fewer outcomes, but we lose the advantage of having things be equally likely to occur.