THE AXIOMATIC APPOACH Let for every event A, a real number P(A) be assigned. Then, P(A) is the probability of event A, if the following axioms are satisfied. Axiom 1: P(A) ≥0 Axiom 2: P(S) = 1, S beings the sure event Axiom 3: For two mutually excusive events A and B
ADDITION THEOREM PROBABILITY For two events A and B, Show that Solution: For events A and B, ---Result 1 Here, A∩B and A`∩B are mutually exclusive. Therefore, by axiom 3, A B S Contd
Also, By result 1 and result 2 -------Result 2 Here, A∩B and A`∩B are mutually exclusive therefore, ADDITION THEOREM PROBABILITY
PERMUTATION& COMBINATION Problems:1. There are 20 persons. 5 of them are graduates. 3 persons are randomly selected from these 20 persons. Find the probability that at least one of the selected person is graduate. ANS 0.6
PROBLEMS:2. In a college, there are five lecturers. Among them, three are doctorates. If a committee consisting three lecturers is formed , what is the probability that at least two of them are doctorates ? ANS 0.7
PROBLEMS:3 What is the probability that there will be 53 Sundays in a randomly selected (i) leap year (ii) non-leap year? ANS P[ leap year has 53 Sundays]= 2/7 P[ non-leap year has 53 Sundays}= 1/7
CONDITIONAL PROBABILITY Let P(A)>0. Then, conditional probability of event B given A is defined as - If P(A) = 0, the conditional probability is not defined.
INDEPENDENT EVENTS Two events A and B are independent if and only if
MULTIPLICATION THEOREM Let A and B be two events with respective probabilities P(A) and P(B). Let P(B/A) be the conditional probability of event B given that event A has happened. Then, the probability of simultaneous occurrence of A and B is – If the events are independent, the statement reduces to - Contd
MULTIPLICATION THEOREM Proof : By the definition of conditional probability, for P(A)>0, If A and B are independent, by the definition of independence,
PROBLEMS:1 The probability that a contractor will get a plumbing contract is 2/3 and probability that he will not get an electrical contract is 5/9. If the probability of getting at least one of these contracts is 4/5, what is the probability that he will get both? ANS 0.1352
PROBLEMS:2 A problem in statistics is given to five students. A, B, C, D and E. Their chances of solving it are 1/2, 1/3, 1/4, 1/5 and 1/6. What is the probability that the problem will be solved?
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