Probability
      Probability definition, terms and notation
         Sample space, event
      Probability formula
      The probability of mutually exclusive events
      The probability of not mutually exclusive events
      Calculating probabilities, examples
Probability definition, terms and notation
The probability is defined as a measure of the degree of confidence in the occurrence of an event, measured on a scale from 0 (impossibility) to 1 (certainty), expressed as the ratio of favorable outcomes of the event to the total number of possible outcomes.
Thus, the probability of an event A occurring,
   
where, 0 <  P(A< 1, so that, P(not A) = 1 - P(A) and P(A) = 1 - P(not A).  
The set of all possible outcomes called the sample space, S = A U not A.  
The probability of mutually exclusive events
The probability of mutually exclusive events, either A or B (or both) occurring is
  P( A U B ) = P(A) + P(B)  
The probability of either events A or B (or both), occurring is written P( A U B ).
The probability of not mutually exclusive events
The probability of not mutually exclusive events, either A or B (or both) occurring is
  P( A U B ) = P(A) + P(B) - P( A B )  
The probability of events A and B both occurring is written P( A B ).
Calculating probabilities, examples
Example:   What is the probability when rolling a die to get the result divisible by 2 or 3? 
Solution:  The probability that an event E will occur,
If rolling a die we get numbers, 2, 4 or 6, fulfilled is condition for the event A, and if we get numbers, 3 or 6, fulfilled is condition for the event B.
Notice that number 6 satisfy both events, so this outcome we should include in the event A or B, thus "or probability of mutually exclusive events
Example:   By rolling three dice at once find probability that the product of all three numbers that come up is divisible by 50.
Solution:  The required condition (the product of all three numbers should be divisible by 50) will satisfy triples, events when three dice show up:  6, 5, 5 and 5, 5, 2.
To calculate the number of favorable outcomes, each of the triples brings, use the formula for permutations of 
n elements (objects), some groups, r, s, t, . . . of which, are the same,
So, each triple ( 6, 5, 5 and 5, 5, 2) gives permutations, where 3 stands for the three shown figures, two 
of which are the same. Therefore, 
where the total number of outcomes 63 equals the number of variations with repetition the three dice, with six sides each, show up. 
Example:   What is the probability that in two throws of a dice the sum of the numbers that come up is 5 or product is 4? 
Solution:  Two throws of a dice we can consider as one throw of two dice. So, the number of favorable outcomes, that the first event occurs (i.e., the sum of the numbers that come up is 5) is determined by pairs,  the event E1:  (1, 4), (2, 3), (4, 1) and (3, 2).
The second event (product is 4) is determined by pairs, and the event  E2:  (1, 4), (2, 2), (4, 1).
Notice that pairs (1, 4) and (4, 1) appear in both events, so we should include them in the event E1 or E2.
Example:  What is the probability that in two throws of a dice the sum of the numbers that come up is 7 or product is 10?
Solution:  The number of favorable outcomes that:   - the event E1 (the sum of the numbers is 7) occurs is determined by pairs:  (1, 6), (2, 5), (3, 4), (6, 1), (5, 2), (4, 3),
     - the event E2 (the product of the shown numbers is 10) occurs is determined by pairs: (2, 5) and (5, 2).
Since the pairs (2, 5) and (5, 2) are already contained in the event E1 then, the probability is
We get the same result by counting the two pairs in the event E2 (but then, we don't count them in the E1), so
Example:  In a box there are 9 balls numbered from 1 to 9. If we draw from the box two balls at once, what is the probability that the sum of both numbers is odd and less than 8?
Solution:  According to stated conditions about the number of favorable outcomes m will give us the following pairs of numbered balls:  (1, 2), (1, 4), (1, 6), (2, 3), (2, 5), and (3, 4).
The total number of possible outcomes is equal to the number of the combinations of the subset with k = 2 elements out of the set of n = 9 elements, i.e.,
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