Definition of 'event' & Random Variables with multiple outcomes - probability

Question

In my introduction to probability course, it says that a sample space is a set of outcomes for a scenario or experiment, such that one and only one outcome will occur. And that an event is simply a subset of these outcomes. Now, pick a real world scenario and say you want to do some probability & statistics investigations on it, and so you want to analyse it so it fits in with the above context.

E.g. roll a regular dice and your event will simply be your score, and it will be only one of {1,2,3,4,5,6}. But you could also have your events being {prime, not a prime}, {3 or 5, neither}, {2,3,4,other} etc. All mutually exclusive outcomes (and so only one from your event will happen), that work with the above definition.

But what if you had a game of rolling a dice, where if you land {1,2,3} Player 1 gets a point, but if you roll {4,5,6} Player 2 & Player 1 both get a point? Now I'm confused as to whether this is an 'event' or not, as defined above (though my instincts tell me it is, under the right circumstances).

For example, if you were to try and define an event of outcomes as {Player 1 scores, Player 2 scores} this would of course not be an event - because one or both of my outcomes could happen. But if you define an event for this game as being {Player 1 scores, both score} that would work perfectly - one and only one of those two outcomes will occur.

This motivates me to conclude then that given a real world scenario, it's your particular analysis that may or may not hold up to the definition of an event, a sample space, an outcome, etc, rather than strictly the scenario itself? Is this all correct - is my take on what are/aren't events in the above scenarios right, or am I missing something?

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This brings me now to the linked problem with the definition of random variables. My introductory course tells me a random variable X is a function from the sample space to some set.

Let me redefine the above scenario in this fashion: say I defined the score of 1-6 on my dice as my sample space, and X maps {1,2,3} and {4,5,6} to {Player 1 scores, both score} respectively. Now again, in this case everything works fine - we've got a bijection between my two events and two final scores, thus X is a function, and so a r.v.

But what if I had the exact same scenario but now defined X as mapping {1,2,3} with 'Player 1 scores' and {4,5,6} with both 'Player 1 scores' & 'Player 2 scores' in {Player 1 scores, Player 2 scores}.

Now, with the exact same setup, and the same players scoring the same points, X is a many-one variable, so not a function and not a random variable.

I conclude again that if you want to do a load of probability and statistics calculations on a scenario, it's not necessarily the scenario itself that may not fit the definition of an event/r.v./etc (though I'm sure there are many real world situations that are undefineable in this way) and be unexaminable, but rather your interpretation of the scenario?

Have I understood correctly what is/isn't an 'event', a random variable, etc, and how their definitions work? Thanks so much!

Really appreciate it.

Answer 1

I would say your example sample space was $\{1,2,3,4,5,6\}$ with each element being an outcome. Then events correspond to elements of a sigma algebra on subsets of the sample space, which with a finite sample space can be all subsets of the sample space

• Player 1 getting a point is an event corresponding to $\{1,2,3,4,5,6\}$
• Player 2 getting a point is an event corresponding to $\{4,5,6\}$
• Both getting a point is an event corresponding to $\{1,2,3,4,5,6\} \cap \{4,5,6\} =\{4,5,6\}$

• Player 1 getting a point and player 2 not is an event corresponding to $\{1,2,3,4,5,6\} \backslash \{4,5,6\} = \{1,2,3\}$

• Neither getting a point is an event corresponding to $\{\}$

Some of these events are mutually exclusive and together span the sample space, such as the third and fourth, or the first and fifth

Answer 2

A random variable is a function from the sample space to a measurable set $S$, not just any set. As a starting point, we typically take $S$ to be the set of real numbers $\mathbb{R}$. So the way you defined the random variable is incorrect.

What you could do, for example, would be to consider $2$ random variables $X_1$, player 1's score and $X_2$, player 2's score. Depending on the outcome of the die roll, they could take the value $0$ or $1$. Of course, the way you define what random variable to consider depends on the problem you're trying to tackle. You could just as well consider another random variable $Y$ which is the sum of both players' points. So it would equal $1$ if the roll was either of $\{1,2,3\}$ or $2$ in case of $\{4,5,6\}$.

As for the definition of events, you could define outcomes in terms of the random variable as well: like $\{X_1 = 1, X_2 = 0\}$, $\{X_1 = 1, X_2 = 1\}$, which are mutually exclusive. In such a case, an event could be something like "Player 1 scores a point" or $X_1 = 1$, and such an event would consist of both the above outcomes.