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Question
A chartered accountant applies for a job in two firms X and Y. He estimates that the probability of his being selected in firm X is 0.7 and beig rejected in Y is 0.5 and the probability that atleast one of his applications rejected is 0.6. What is the probability that he will be selected in one of the firms?

Naveen,

Let the event X = selected at company X and the event Y = selected at company Y. This means that X' = rejected at X and Y' = rejected at Y. Here the prime notation means the "complement of". From the information given, we have

P(X) = 0.7
P(Y') = 0.5

from which

P(X') = 0.3
P(Y) =  0.5.

using P(X) = 1 - P(X'), etc.

As stated, we want the probability of the event X∪Y = union of events X and Y =  either X or Y, or both, happens (one or the other or both of the applications being selected). In other words, we want P(X∪Y), which can be written, using the axioms of probability theory, as

P(X∪Y) = P(X) + P(Y) - P(X∩Y)  where the ∩ symbol means the "intersection of" or, in other words, both X and Y happen. So now we need to calculate P(X∩Y).

From the information given, we know that the probability of either X' or Y', or both, happening is 0.6, which can be written

P(X'∪Y') = 0.6.

From the axioms of set theory we have (X'∪Y') = (X∩Y)' = complement of the intersection of X and Y (complement of both X and Y happening). We also have from probability theory that P(event) = 1 - P(event'), so that

P[(X∩Y)'] = P[(X'∪Y')] = 1 - P[(X∩Y)] = 0.6

or

P[(X∩Y)] = 1 - 0.6 = 0.4.

From the equation above

P(X∪Y) = P(X) + P(Y) - P(X∩Y) = 0.7 + 0.5 - 0.4 = 0.8, which is your answer.

Note that this means that the events X and Y are not independent since, if we make that assumption

P(X∪Y) = P(X) + P(Y) - P(X∩Y) = P(X) + P(Y) - P(X)･P(Y) = 0.7 + 0.5 - (0.7)(0.5) = 0.85

which is slightly different.

Hope this makes sense.

Randy
Questioner's Rating
 Rating(1-10) Knowledgeability = 10 Clarity of Response = 10 Politeness = 10 Comment Thanks a lot sir. You did a great help for me!

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#### Randy Patton

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Questions regarding application of mathematical techniques and knowledge of physics and engineering principles to product and services design, optimization, prediction, feasibility and implementation. Examples include sales and product performance projections based on math/physics models in addition to standard regression; practical and cost effective sensor design and component configuration; optimal resource allocation using common tools (eg., MS Office); advanced data analysis techniques and implementation; simulation and "what if" analysis; and innovative applications of remote sensing.

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26 years as professional physical scientist and project manager for elite research company providing academic quality basic and applied research for government and defense industry clients (currently retired). Projects I have been involved in include: - Notional sensor performance predictions for detecting underwater phenomena - Designing and testing guidance algorithms for multi-component system - Statistical analysis of ship tracking data and development of anomaly detector - Deployed vibration sensors in Arctic ice floes; analysis of data - Developed and tested ocean optical instrument to measure particles - Field testing of protoype sonar system - Analysis of synthetic aperture radar system data for ocean surface measurements - Redesigned dust shelters for greeters at Burning Man Festival Project management with responsibility for allocation and monitoriing of staff and equipment resources.

Publications
“A Numerical Model for Low-Frequency Equatorial Dynamics” (with Mark A. Cane), J. of Phys. Oceanogr., 14, No. 12, pp. 18531863, December 1984.

Education/Credentials
MIT, MS Physical Oceanography, 1981 UC Berkeley, BS Applied Math, 1976

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