An integrated circuit factory has three machines X, Y, and Z. Test one integrate

Question

An integrated circuit factory has three machines X, Y, and Z. Test one integrated circuit produced by each machine. Either a circuit is acceptable (a) or it fails (f). An observation is a sequence of three test results corresponding to the circuits from machine X, Y, and Z, respectively. For example, aaf is the observation that the circuits from X and Y pass the test and the circuit from Z fails the test. (a) What are the elements of the sample space of this experiment? (b) What are the elements of the sets: Z_F = {circuit from Z fails} X_A = {circuit from X is acceptable} (c) Are Z_F and X_A mutually exclusive? (Explain) (d) Are Z_F and X_A collectively exhaustive? (Explain) (e) What are the elements of the sets: C = {more than one circuit is acceptable} D = {at least two circuits fail} (f) Are C and D mutually exclusive? (Explain) (g) Are C and D collectively exhaustive? (Explain)

Explanation / Answer

(a) The elements of the sample space are the following:

{aaa, faa, afa, aaf, ffa, faf, aff, fff}

(b) elements of Zf are {aaf, faf, aff, fff}

elements of Xa are {aaa, afa, aaf, aff}

(c) Zf and Xa are not mutually exclusive because the following elements are common to both {aaf, aff}

(d) Zf and Xa are not collectively exhaustive because the following elements are absent from both of them {faa, ffa}

(e) Elements of C are based on acceptability of more than one circuit (i.e 2 or 3 circuits)

Thus, elements of C are {aaa, faa, afa, aaf}

Elements of D are based on failure of atleast two circuits (i.e 2 or 3 circuits)

Thus, elements of D are {ffa, faf, aff, fff}

(f) C and D are mutually exclusive because none of the elements are common between them

This is because amongst three machines, if more than one circuit is acceptable, atleast two circuits cannot fail

(f) C and D are collectively exhaustive because the elements of both together form the elements of the sample space. There is no elements which they collectively do not cover.

This is because amongst three machines, if one circuit is acceptable, atleast two circuits will fail.

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