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


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.

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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Randy Patton


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