Probability

Probability : Probability starts with a sample space that describes possible outcomes in an experiment. Probability consists of sample space which contain all the elements and from these elements we select the outcome required for the experiment to get the probability.

Examples : 

1. Coin Tossing :

                               AGAINST ALL ODDS… an exercise in GRATITUDE – Jim Morouse 
 S = {H,T}
it gives us idea for the one toss where S is the Sample space and {H,T} are the outcomes. If the coin is tossed twice the possible sample  space can be :
                              S = {HH,HT,TH,TT}
For multiple tosses it could be :
                             S = {HH .........H, HH........HT , HH.........HTH,.........,T......T}

2. Roll Die :
                                Discussion 3: Rolling the Dice
If Single die is rolled then the sample space can be :
                                 S = {1,2,3,4,5,6}
If two dices are rolled then the sample space can be :
                                 S = {11,12,13,.....,16,21,22,23,.........,66}

Sample Spaces are sets. If S is a sample space and A C S (A is subset of S), we can call A an event, it is basically the condition on which the experiment calculate the probability for that particular event.

Examples:

1. A = {Three heads when tossing a coin six times }
                     A = {HHHTTT, HHTHTT, HHTTHT,...............,TTTHHH}

2. A = {Striaght flush for poker hands }
                   A = {AS KS QS JS 10S, KS QS JS 10S 9S,.............,5C 4C 3C 2C AC}
3. In an experiment with two outcomes , S and F where S is Success and F is Failure which is performed n times.
A = {k Successes were observed}
                A = {1,2,3,4,...........,n}
If A and B are events :

  •  A U B : It is Union where it contains all the elements present in A or B.                                            P(A U B) = P(A) + P(B)                                                                A U B = {x belongs to S : x belongs to A or x belongs to B}
  • A ∩ B : It is intersection where it contains all the elements that are common among A and B.                                                                                                                        A ∩ B = {x belongs to S : x belongs to A and x belongs to B}
  • A \ B : It is difference where it contains all the elements are present in A but not in B.                                                                                                                                         A / B = {x belongs to S : x belongs to A and x does not belongs to B} 
  •  = { x belongs to S : x does not belong to A}


A probability is an assignment of a value to each event in a sample space , the value must lie in [0,1] and is P(A) denotes the value assignment to A and A1 , A2, ......,An are mutually disjoint events where A ∩ A = ∅           

Counting : This gives a way to compute probabilities of events when the probability of each outcome is same. If S is a sample space with probability P such that P(w)  = P(w1) for all w , w1 belongs to S.
if A C S then P(A) = #A / #S where #A  is the number of elements of A and #S is the number of elements of S

Multiplication Principle : If experiment 1 has m possible outcomes and experiment 2 has n possible outcomes then 1  and 2 has m.n possible outcomes.

Example:

If I toss a coin then roll a die , how many outcomes are there?
          outcomes for coin(m) = 2 (Head and tail)
          Outcomes for die(n) = 6 (1,2,3,4,5,6)
                      m.n = 2.6 = 12
Permutation: A permutation is an ordering at distinguishable objects.

Example: How many ways can a committee of size 5 be selected from a group of 47 without replacement?
                                   P(A) = 47*46*45*44*43 / 5!
                                       

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