Question

Consider a manufacturing line with 3 robotic machines used to create a widget. Each robotic machine either performs (P) perfectly or creates a (D) defect. Denote the outcome of the widget by creating 3-tuples of P's and D's.

1) Find the sample space S.

2) Find the event A that at least one robotic machine created a defect (D).

3) Find event B that all three robotic machines performed the same way.

4) Are the events A and B mutually exclusive? Show mathematically using set theory notation.

Answer 1

A robotic machine performs perfectly is denoted by P

A robotic machine creates a defect is given by D

There are 3 robotic machines

The sample space will contain 2³ =8 elements

The sample space is given by

s={PPP,PPD,PDP,PDD,DPP,DPD,DDP,DDD}

A be the event that at least one robotic machine created a defect

A={PPD,PDP,PDD,DPP,DPD,DDP,DDD}

n(A)=no. of cases favourable to the event=7

n=no. of possible outcomes in the sample space s=8

so P(A)= probability that at least one robotic machine created a defect=n(A)/n=7/8

now B be the even t that all three robotic machines performed the same way

B={PPP,DDD}

n(B)=no. of cases favourable to the event=2

P(B)=probability that all the machines performed in the same way=n(B)/n=2/8=.25

now we know that two events are mutually exclusive if

P(A$\cap$B)=P(A)*P(B)

Now A$\cap$B= {DDD}

n(A$\cap$B)=no. of cases favourable to the event A$\cap$B=1

P(A$\cap$B)=n(A$\cap$B)/n=1/8

P(A)*P(B)=7/8*2/8=14/64

so P(A$\cap$B)$\neq$P(A)*P(B)

so the events A and B are not mutually exclusive.

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