Problem 4. [2+6+3+3+3+14-31 pointsl The efficiency of a UV disinfectant is under

Question

Problem 4. [2+6+3+3+3+14-31 pointsl The efficiency of a UV disinfectant is under investigation. Three water samples are taken from a lake and treated in succession. Let C denote the event that a sample is treated successfully (i.e. it conforms with the existing drinking water quality standards), so that C is the event that a sample is treated unsuccessfully. Thus, for example, one possible outcome is CCC, which means that the first and last samples are treated successfully while the second is not.

Explanation / Answer

Part (a)

When 3 samples are treated there are 8 possible outcomes, since each treatment can be successful or unsuccessful. Let Ci represent outcome that ith treatment is successful which in turn implies CiC represents the outcome that ith treatment is not successful, i = 1, 2, 3.

Then, the sample space is:

{(C1C2C3), (C1C2C3C), (C1C2CC3), (C1CC2C3), (C1CC2C C3), (C1CC2C3C), (C1C2CC3C), (C1C C2C C3C)} ANSWER

Part (b)

A1

Complement of ‘at least one sample treated is successful’ is ‘No sample treated is successful’.

Thus, elements that constitute ‘at least one sample treated is successful’ are:

{(C1C2C3), (C1C2C3C), (C1C2CC3), (C1CC2C3), (C1CC2C C3), (C1CC2C3C), (C1C2CC3C)} ANSWER

A2

Element that constitutes ‘all samples treated are successful’ is: (C1C2C3) ANSWER

A3

Elements that constitute ‘only one sample treated is successful’ are:

{ (C1CC2C C3), (C1CC2C3C), (C1C2CC3C)} ANSWER

Part (c)

A1 and A2 are not disjoint ANSWER because (C1C2C3) is common to both.

Part (d)

A1 and A3 are not disjoint ANSWER because { (C1CC2C C3), (C1CC2C3C), (C1C2CC3C)} are common to both.

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