Sample spaces, events and set theory intersection

1. Jan 15, 2015

s3a

1. The problem statement, all variables and given/known data
Problem:
Given a regular deck of 52 cards, let A be the event {king is drawn} or simply {king} and B the event {club is drawn} or simply {club}.

Describe the event A ∪ B

Solution:
A ∪ B = {either king or club or both (where "both" means "king of clubs")}

2. Relevant equations
Set theory intersection.

3. The attempt at a solution
I just wanted to ask whether A and be are subsets of different sample spaces or not. Is there one sample space for the suits and one sample space for the types of cards per suit? In other words, are sets A and B subsets of different sample spaces?

I ask because, if I think of A and B being subsets of the same sample space, then I can think of the sample space either being

$S_1$ = {A♥, A♠, A♦, A♣, 2♥, 2♠, 2♦, 2♣, 3♥, 3♠, 3♦, 3♣, 4♥, 4♠, 4♦, 4♣, 5♥, 5♠, 5♦, 5♣, 6♥, 6♠, 6♦, 6♣, 7♥, 7♠, 7♦, 7♣, 8♥, 8♠, 8♦, 8♣, 9♥, 9♠, 9♦, 9♣, 10♥, 10♠, 10♦, 10♣, J♥, J♠, J♦, J♣, Q♥, Q♠, Q♦, Q♣, K♥, K♠, K♦, K♣}

or

$S_2$ = {(A,♥), (A,♠), (A,♦), (A,♣), (2,♥), (2,♠), (2,♦), (2,♣), (3,♥), (3,♠), (3,♦), (3,♣), (4,♥), (4,♠), (4,♦), (4,♣), (5,♥), (5,♠), (5,♦), (5,♣), (6,♥), (6,♠), (6,♦), (6,♣), (7,♥), (7,♠), (7,♦), (7,♣), (8,♥), (8,♠), (8,♦), (8,♣), (9,♥), (9,♠), (9,♦), (9,♣), (10,♥), (10,♠), (10,♦), (10,♣), (J,♥), (J,♠), (J,♦), (J,♣), (Q,♥), (Q,♠), (Q,♦), (Q,♣), (K,♥), (K,♠), (K,♦), (K,♣)}

(or the same kinds of sets using different symbols).

Neither $S_1$ nor $S_2$ have subsets that are {king} = {K} or or {club} = {C}.

Could someone please clarify this for me?

2. Jan 15, 2015

Staff: Mentor

For the event (a king is drawn) you ignore the suit, so any of the four kings would be included in this event. For the event (a club is drawn), you ignore the rank, and any of the 13 cards in this suit would be included.

3. Jan 15, 2015

LCKurtz

You are confusing an English description of the event with the event itself. In either case above, you have sample spaces that contain a representation of the 52 card deck. When you talk about the event that a king is drawn you are describing a subset of the sample space, which would be $\{K♥, K♠, K♦, K♣\}$ or $\{(K,♥), (K,♠), (K,♦), (K,♣)\}$. There is no event {king}. Remember, events are subsets of the sample space, however you describe them in a sentence.

4. Jan 15, 2015

s3a

LCKurtz, I agree that the sets you gave are valid events assuming $S_1$ and $S_2$ are the sample spaces worked with.

The thing is, the book says {king} and {club} and does not mention the sample space sets $S_1$ and $S_2$ that I mentioned above. Given that the book claims that {king} and {club} are events, what is a (correct) sample space that is a superset of the {king} and {club} events? Is there any, or did the book make a mistake?

Mark44, when I try to be mathematically rigorous, what you're saying doesn't sit well with me, because neither {K} (K for king) nor {C} (C for club) is a subset of either $S_1$ or $S_2$.

5. Jan 16, 2015

LCKurtz

The only way that makes sense is if you name the subsets king and club. For example:

king = $\{K♥, K♠, K♦, K♣\}$ and club similarly.

6. Jan 16, 2015

Ray Vickson

Strictly speaking, an event is a subset of the sample space so should not have parentheses around its name. Thus, we have (for example)
$$\rm{King} = \{ \text{King of hearts, King of spades, King of clubs, King of diamonds} \}\\ = \{ \rm{(KH),(KS),(KC),(KD)} \}.$$
Here, the parentheses are around the elements of the subset, not around the name of the subset.

However, when writing things out it might be that you want to use the word "King" or "Kings" in two different ways, one as some type of general description and one as the name of an event in the sample space. Since we want to avoid mixing up the meanings of these two usages, it is useful to put parentheses around the word when we want to make it clear it is a subset, so we could write {King} for the event and 'King' for some other type of usage.