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PROBABILITY

PROBABILITY It is a measure of the likelihood of a random phenomenon or chance behavior. Probability describes the long-term proportion with which a certain outcome will occur in situations with short-term uncertainty. EXAMPLE:

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PROBABILITY

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  1. PROBABILITY It is a measure of the likelihood of a random phenomenon or chance behavior. Probability describes the long-term proportion with which a certain outcome will occur in situations with short-term uncertainty. EXAMPLE: Simulate flipping a coin 100 times. Plot the proportion of heads against the number of flips. Repeat the simulation.

  2. THEOREMS OF PROBABILITY ADDITION THEOREM MULTIPLICATION THEOREM

  3. Addition Rule for Mutually Exclusive Events • If E and F are mutually exclusive events, then • P(E or F) = P(E) + P(F) • In general, if E, F, G, … are mutually exclusive events, then • P(E or F or G or …) = P(E) + P(F) + P(G) + …

  4. Given two mutually exclusive events A and B such and , find P(A or B). Example 1 :

  5. An integer is chosen at random from the first 200 positive integers. Find the probability that the integer is divisible by 6 or 8. Solution: Let S be sample space. Then, S = {1, 2, 3, …200}, n(S) = 200 Let A : event that the number is divisible by 6. A = {6, 12, 18 ... 198}, n(A) = 33 Example:2 Let B : event that number is divisible by 8. B = {8, 16, 24 ... 200}, n(B) = 25

  6. (A Ç B) : event that the number is divisible by 6 and 8. A Ç B = {24, 48, ... 192}, n(A Ç B) = 8 A  B : event that the number is divisible either by 6 or 8. SolutionCont.

  7. For finding the probability of one or more of two events that are not mutually exclusive the modified addition theorem is used: P(A or B) = P(A) + P(B) – P(A and B) Where P(A or B) = Probability of happening of A and B when A and B are not mutually exclusive. P(A) = Probability of happening of event A. P(B) = Probability of happening of event B. P(AB) = Probability of happening of events A and B together in case of three events P(A or B or C) = P(A) + P(B) +P(C) – P(AB) – P(AC) – P(BC) + P( ABC) When Events are not mutually Exclusive

  8. MULTIPLICATION THEOREM INDEPENDENT EVENTS: Two events E and F are independent if the occurrence of event E in a probability experiment does not affect the probability of event F. Two events are dependent if the occurrence of event E in a probability experiment affects the probability of event F. DEFINITION OF INDEPENDENT EVENTS: Two events E and F are independent if and only if P(F | E) = P(F) or P(E | F) = P(E)

  9. The probabilities of A, B and C solving a problem are respectively. If the problem is attempted by all simultaneously, find the probability of exactly one of them solving it. Example :

  10. Required probability [As A, B’ and C’ are independent events; A’, B and C’ are independent events; A’, B’ and C are also independent events] Solution ( Cont. )

  11. In a random experiment, if A and B are two events, then the probability of occurrence of event A when event B has already occurred and , is called the conditional probability and it is denoted by MULTIPLICATION THEOREM IN CASE OF CONDITIONAL PROBABILITY:

  12. Two events are independent if the occurrence of one of the events does NOT affect the probability of the occurrence of the other event. Two events A and B are independent if: P(B|A) = P(B) or if P(A|B) = P(A) Events that are not independent are dependent Independent and Dependent Events

  13. Q) Two cards are selected without replacement, from a standard deck. Find the probability of selecting a king and then selecting a queen. Solution: Because the first card is not replaced, the events are dependent. P(K and Q) = P(K) ● P(Q|K) So the probability of selecting a king and then a queen is about .0006 EXAMPLE :

  14. Thank You

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