Summary of Basics of probability and Algebra of Events in 10 points:
- Any experiment that has more than one possible outcome and its outcome cannot be predicted in advance is called random experiment.
For example, tossing of a coin. - The total possible outcomes of any random experiment is called the sample space of the experiment (denoted by S) and each outcome is called the sample point.
Example: Two coins are tossed, then the Sample Space = {HH, HT, TH, TT} and 4 outcomes will be the 4 sample points which make the sample space.
(NOTE: We take the sample points as ordered pairs. So, HT and TH will be two sample points.)
Example: A coin is tossed. If head shows up, then a dice is thrown. If tail shows up, a candy is drawn from the bad of 5 brown and 2 yellow candies. Now, Sample space = {H1, H2, H3, H4, H5, H6, TB1, TB2, TB3, TB4, TB5, TY1, TY2} and there are total 13 sample points. - Any event is the subset of the sample space.
As in example, the Sample Space (S) = {HH, HT, TH, TT}. Now, we want that only those outcomes that contain atleast one head, then the set E = {HT,TH,HH}. This set E is an event.
Clearly, all the elements of the event E are in the sample space. So, set E is the subset of sample space S i.e. E ⊂ S. - The set which does not contain any element is called an empty set. It is denoted by Φ. Also, called as null set.
- The impossible even is the empty set and a sure event is the set that always contain atleast one sample point.
Example: When a dice is thrown, Sample space = { 1,2,3,4,5,6,}
Then, Set(Φ) = { Any number greater than eight } = {0}.This is a null set.
Set(E) = { The set of even number } = {2,4,6}.This is a sure set. - The set that contains exactly one element from the sample space is called a simple event.
Example : When two coins are tossed, then the Sample Space = {HH, HT, TH, TT}. The set that contains exactly two heads, i.e. Set (E) = {HH}. This is a simple event. - The set that contains more than one element from the sample space is called a compound event.
For the Sample Space (S) = {HH, HT, TH, TT}. Now, we want that only those outcomes that contain atleast one head, then the set E = {HT,TH,HH}. This set E is a compound event. - The EVENT "A" OR "B" : This shows the union of two sets. It contains elements from either set A or set B. The union of set is denoted as A U B
Example : Set (A) = {a,b,c,d,e,f,g,h,i,j,k,l,m,n} and Set (B) = {a,e,i,o,u}
Then, Set (A U B) = {a,b,c,d,e,f,g,h,i,j,k,l,m,n, a,e,i,o,u} - The EVENT "A" AND "B" : This shows the intersection of two sets. It contains only those elements which are common in both the sets A and B. The intersection of sets is denoted as A ∩ B.
Example : Set (A) = {a,b,c,d,e,f,g,h,i,j,k,l,m,n} and Set (B) = {a,e,i,o,u}
Then, Set (A ∩ B) = {a,e,i} - The EVENT "A" BUT NOT "B" : This shows the those elements which are present in set A but not in set B. It is denoted as A - B.
Example : Set (A) = {a,b,c,d,e,f,g,h,i,j,k,l,m,n} and Set (B) = {a,e,i,o,u}
Then, Set (A - B) = {b,c,d,f,g,h,j,k,l,m,n}
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