BUSINESS MATHEMATICS & STATISTICS

Elementary Probability LECTURE 35 Review Lecture 35 Elementary Probability Part 2

PROBABILITY Making assessment of chances A worker out of 600 gets a prize by lottery Chance of any one individual say Rashid being selected = 1/600 The probability of the event ”Rashid is selected” is the probability of an event occurring P(Rashid = 1/600) This is a priori method of finding probability as we can assess the probability before the event occurred

PROBABILITY When all outcomes are equally likely a priori probability is defined as: P(event) = Number of ways that event can occur/Total number of possible outcomes If out of 600 persons 250 are women, then the chance of a women being selected = p(woman) = 250/600

PROBABILITY In many situations there is no prior knowledge to calculate probabilities What is the probability of a machine being defective? Method: Monitor the nmachine over a period of time Find out how many times it becomes defective This experimental or empirical approach

PROBABILITY P(event) = Number of times event occurs/Total number of experiments Larger the number of experiments more accurate the estimate Experimental probability approaches theoretical probability as the number of experiments becomes very large

PROBABILITY There are two events A and B What is the probability of either A or B happening? What is the probability of A and B happening? Number of possibilities Probability of A or B hapenning = Number of ways A or B can happen/ Total number of possibilities = Number of ways A can happen + number of ways B can happen/ Total number of possibilities

PROBABILITY = Number of ways A can happen/ Total number of possibilities + Number of ways B can happen/ Total number of possibilities = Probability of A happening + Probability of B happening Condition A and B must be mutually exclusive When A and B are mutually exclusive p(A or B) = p(A) + p(B)

EXAMPLE If a dice is thrown what is the chance of getting an even number or a number divisble by three? P(even) = 3/6 p(div by 3) = 2/6 p(even or div by 3) = 3/6 + 2/6 = 5/6 The number 6 is not mutually exclusive Hence correct answer = 4/6

AND RULE Probability of A and B happening = Probability of A x Probability of B Example 40% workforce are women p(woman chosen) = 2/5 25% females = management grade 30% of males = management grade What is the probability that a worker selected is a women from management grade?

EXAMPLE p(woman & Management grade) = p(woman) x p(management) Total workforce = 100 p(woman) = 0.4 p( management) = 0.25 p(woman) x p( management) = 0.4 x 0.25 = 0.1 or 10%

SET OF MUTUALLY EXCLUSIVE EVENTS Between them cover all possibilities Probabilities of all these events together add up to 1 p(A) + p(B) + p(C) +....p(N) = 1 Exhaustive Events A happens or A does not happen p(A happens) + A (does not happen) = 1 Example p(you pass) = 0.9 p(you fail) = 1 – 0.9 = 0.1

EXAMPLE A production line uses 3 machines Chance that 1st machine breaks down in any week = 1/10 Chance for 2nd machine = 1/20 Chance of 3rd machine = 1/40 What is the chance that at least one machine breaks down in any week?

EXAMPLE P(at least one not working) + p(all three working) = 1 p(all three working) = p(1st working) x p(2nd working) x p(3rd working) p(1st working) = 1 - p(1st not working) = 1- 1/10 = 9/10 p(2nd working) = 19/20 P(3rd working) = 39/40 P(all working) 9/10 x 19/20 x 39/40 = 6669/8000 P(at least 1 working) = 1- 6669/8000 = 1331/8000

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