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
