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Math and Science Solutions for Businesses/Maths [ 10th grade probability ]

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Question
A problem in maths is given to three students whose chances of solving it are respectively 1/2 , 1/3 and 1/4. What is the probability that the problem will be solved ?
        
Please explain in such a way that a 10th grade student can understand.

Answer
This a classic example of how to use a very useful shortcut in probabilty problems, namely

(Probability something will happen) = 1 - (Probability something won't happen),

which for this particular problem is

(Probability problem will get solved) = 1 - (Probability the problem won't get solved).

The probability on the right hand side is easier to calculate, as follows:

The outcome of the student solving the problem or not is called an event, and each student's attempt represents an independent event. We need to combine the probabilities of the individual students to calculate the overall probability of the problem being solved. This can happen in more than one way, namely 1 of the students solves it but the others do not, 2 students solve it but one doesn't, etc. This obviously gets complicated, though the algebra of sets and probabilities can give you an explicit expression.

An easier way to approach the solution is to note that there is only one way that the math problem isn't solved, namely if all 3 students don't solve it. Since these events are independent, the total probability is the product of the individual probabilities, i.e.,

Probability problem won't get solved = (prob student 1 doesn't solve it)・(prob student 2 doesn't)・(prob student 3 doesn't).

Now we can again apply the rule above to calculate these individual probablities,

(prob student n doesn't solve problem) = 1 - (prob student n does), or P_n_no = 1 - P_n_yes, or

P_1_no = 1 - 1/2 = 1/2
P_2_no = 1 - 1/3 = 2/3
P_3_no = 1 - 1/4 = 3/4

so that

Probability of problem being solved = 1 - (1/2)・(2/3)・(3/4) = 1 - 0.25 = 0.75

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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.

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

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

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