Question

A lightbulb manufacturer ships large consignments of lightbulbs to big industrial users. When the production process is functioning​ correctly, which is 60 % of the​ time, 25​% of all bulbs produced are defective.​ However, the process is susceptible to an occasional​ malfunction, leading to a defective rate of 50​%. If a defective bulb is​ found, what is the probability that the process is functioning​ correctly? If a nondefective bulb is​ found, what is the probability that the process is operating​ correctly?

Answer 1

Answer:

a) P[ Pf | Ptd ]  = 0,428

b) P[ Pf | Nd ]  = 0,692

Step-by-step explanation:

We will call:

Pf  Probability of functioning correctly    60 %   or 0.6

Pfn  Probability of malfunctioning   40 %  or  0,4

PNd Probability of nondefective bulb

D₁  Probability of defective bulb when the process is correctly working

D₂  Probability of defective bulb when the process is in malfunction condition

Ptd Total probability of defective bulbs

Then applying theorem´s Bayes  have

P [ A | B ]  =  P(A) * P [ B | A ] / P(B)

a) Probabilty of the process is functioning correctly given that we pick up a defective bulb

The total probability of defective bulbs is equal to, probabilty of defective bulbs when the process is Ok, plus probability of defective bulbs when the process  is in malfunctioning condition, therefore

Ptd  = (0,6)*(0,25) + (0,40)*(0,50) = 0,15 * 0,20 = 0,35

P[ Pf | Ptd ]  =  Pf * P [ Pd | Pf ] / Ptd    =  0,6 * 0,25 / 0,35

P[ Pf | Ptd ]  = 0,428

b)  P [ Pf | Nd ]

P[ Pf | Nd ]  =  Pf *  P [Nd | Pf ] / PNd

PNd  = 1 - Ptd    =  1 - 0,35   =  0,65

P[ Pf | Nd ]  = 0,6 * 0,75 / 0,65   =  0,692

P[ Pf | Nd ]  = 0,692

Answer 2

A) The probability that the process is functioning​ correctly if a defective bulb is found = 0.428

B) The probability that the process is operating​ correctly if a non-defective bulb is found = 0.692

What is the Conditional Probability?

We are given:

Probability of functioning correctly; P_fc = 60% = 0.6

Probability of malfunctioning P_nfc = 40% = 0.4

Probability of nondefective bulb P(nd)

Probability of defective bulbs when the process is Okay = 0.6 × 0.25 = 0.15

Probability of defective bulbs when the process  is in malfunctioning condition = 0.4 × 0.5 = 0.20

Then using Bayes theorem, we have;

a) Total probability of defective bulbs is equal to is;

P_td  = (0.6 × 0.25) + (0.4 × 0.5)

P_td = 0.15 + 0.20 = 0.35

Thus, probability that the process is functioning correctly is;

P[P_fc | P_td]  =  P_fc × P[P_d|P_f ]/P_td

P[P_fc | P_td] =  (0.6 × 0.25)/0.35

P[P_fc | P_td] = 0.428

B)  We want to find the the probability that the process is operating​ correctly If a non-defective bulb is​ found. Thus;

P[ P_fc|Nd ]  =  P_fc ×  P[Nd | Pf ]/P_Nd

Now;

P_Nd  = 1 - P_td    

P_Nd =  1 - 0.35  

P_Nd =  0.65

P[P_fc|Nd]  = (0.6 × 0.75)/0.65  

P[P_fc|Nd]  = 0.692

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