Chapter 2: basic counting and probability

                                         Basic counting and probability

Let’s count the number of words of length  K with n letters  ( repetition is  authorized): So  we have   A ,  B , B ,A ,  C ,   (K = 5    n= 3 )   generally   we  can wright (n exponant k ) words.

Now let’s count words from alphabet of k letters with repetion not allowed: K= 5  this kind of counting is called permutation of k objects :

A  B  C   D   E ( in this kind of counting each letter can be used only once So we will have 5 choices for the first letter , 4 for the second ,3 for the third ,2 for the forth , and 1 for the fifth .

K (K-1)(k-2)(k-3)(k-4) 1 =K!

Now let’s count words of length K from alphabet of n letters :N choices for the first letter,(N-1) choices for the second letter,(N-2 )choices for the third letter……(N-k)+1 choices for k th letter .That means n!/(n-k)! Words and without permutation we must divide by K!   so thefinal formula will be :   n!/(n-k)!k!   (that is the binomial coefficient)So for example for(3 exponant 5 )  we will have 5!/(5-3)!3! = 10   eg : from  A,B,C,D,E  we can have 10 possibilities:

{A,B,C}    {A,B,D}    {A,B,E}   {A,C,D}    {A,C,E}   { A,D,E}    {B,C,D}    {B,C,E}    {B,D,E}   {C,D,E}

                               BASIC DEFINITION OF PROBABILITY

Suppose a certain outcome o from N ways out of T possibilities. this is between 0 and 1    0 <  p <1

Ex:  Rolling a dice the Probability of getting a 3= 1/6  and  in a class of 28 peoples, 4 have red hair. Probability of red hair = 4/28 = 1/7

Ex: Probability of getting 3 successive Heads if you flip a coin 3 times = 1/2 x 1/2 x 1/2 = 1/8

Ex: Rolling tow dices the probability of getting 8 : well in this case the space of

Possibility is( total number is 36 and)  and outcomes giving 8 are 7 so the probability is  = P = 7/36

                                              RULES FOR PROBABILIOTY

Negation Rule: if the probability of an outcome o is P then the probability of O not occurring is (1-P).

Ex: Probability of rolling a 3 is 1/6 so that the probability of not rolling a 3 = 5/6

Multiplication Rule: if O1 and O2 are independent outcomes that means knowing that one does not influence the other. Then the probability of (O1 and O2) occurring together is the Prob(O1) x Prob(O2)

Ex: Wat´s the probability of drowing a red face card (Jack,Queen,King).

Red cards= hearts and diamonds . Wat`s the probability of getting a red card =2/4 and the Probability of getting a face card = 3/13 so the probability of drowing a red face card is 2/4 x 3/13 = 6/52 = 3/2

Addition Rule: if an outcome O can occur in two or more distinct possibles ways, then the probability of O is the  some   of the probabilities of each of individual ways of getting O.

Ex: Probability of getting  2 (3) if we roll a dice 5 times:Probability of getting a 3 in exactly first and forth rolles  is  the same  and not getting 3 in the second rol = 1-4/6 Now the probability of any of two positions   Then final probability =0,161

                                              POCKER CALCULATION

What is the probability of getting at least one pair from of 5 card hand (out of 52) .1-P=probability of getting all cards from different values 1   so   P=1-0,4929=1/2

Common birthdays: What´s the probability of ten people in the room at least two share birthday.Let´s find that  1-P=all ten have distinct birthday1-P=1    so P=1-0,883=0,117   and With 23 people P

      

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