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
ThisisbetweenProbabilityofgettingEx: