Hello,
If the sample space is a "set" then there should be no repetition in its elements right?
So if i come across a problem such as....
"A bag contains two red balls, three green balls and five black balls, calculate the probability of picking up a red ball with first try?"
Here the the probability would be "2/10" , cuz the sample space apparently contains 10 elements.
n(S)=10 S={red ball, red ball, red ball, green ball, green ball, green ball, black ball, black ball, black ball, black ball, black ball}
This set looks wrong... shouldnt it be, by the definition of sample set, n(S)= 3 &
S={ red ball, green ball, black ball}
Thanks
So why does every author in every book use the word SET for the sample space, there is a huge difference between collection and set. Merely saying that sample space is a collection, rather than a set does not solve my problem. I would prefer to have a thorough explanation about this issue.
Probability texts are most often written by mathematicians. Many think that every mathematical concept is about some set.
Suppose that we have a collection (committee, team, class, club) of two English men, three French men, and five German men. This collection is a SET of ten. The bag of ten balls you asked about in the OP is no different in concept from the set of ten men of three nationalities. Both are finite sets of ten members. For the purposes of elementary probability the bag you posted can serve as a model of the team I posted. Any basic probability question about my team has a corresponding question about your bag. The answer is the same in both cases.
Khamaar,
There is nothing wrong with the idea of a sample space being a set, but what you need to keep in mind is what the sample space S contains. In the case of balls, you might want to think of them as distinguished balls. For instance, if you have two red balls, one of them may be labeled '1' and the other '2'. Of course, our probability has no concern for which of these distinguished items was chosen, only that it was red. If you could not distinguish the items, you would be right. The set {r, r} = {r}. It is implicit in the very fact we are saying there are two red balls that there are two distinguished red balls in the set. Some authors do make this point explicit, I might add.
Consider a different experiment consisting of two coins being tossed. Here the sample space consists of events we can intuitively identify as {TT, TH, HT, HH}. This is the same as if we were to toss one coin twice. It does not matter which coin produced the 'T' or the 'H', however. Assuming the outcome of each coin (or toss) is independent of the other, our sample space consists of merely those outcomes of interest. It is implicit that the coins (or tosses) are distinguishable. Now, we could attach a number to identify which coin (or toss) produced the outcome, by saying it was coin 1 or toss 2 that produced the 'H'. But notice that in terms of the sample space and identifying the outcomes (which is what the sample spaces consists off), we have no need for those identifiers. Therefore, we drop them, for to include them would be to ask a different question--viz., we would be asking a question about the outcome for one of the distinguished items, not their place in a larger outcome (e.g., the number of 'T's in two coin tosses). Do you see the difference?