The company will use machine A,B,C,D with probabilities 0.23,0.45,0.19 and 0.13 respectively.From past experience it is known that the probability of defective tablet produced by the machine are 0.11,0.2,0.07 and 0.05 respectively.Suppose a defective tablet is produced： a)what is the probability of machineD? b)what is the probability of machineB?
Apply Bay's rule here
A-p(E4/D)=)0.05*0.13/(0.23*0.11)+(0.45*0.2)+(0.19*0.07)+(0.13*0.05)
=5/196
Similarly you can find for machine B
To find the probabilities in this scenario, you will use conditional probability. Conditional probability is the probability of an event occurring given that another event has already occurred. In this case, the event is producing a defective tablet, and we want to find the probabilities of machines A, B, C, and D given that a defective tablet has been produced.
Let's calculate the probabilities:
a) Probability of machine D given a defective tablet:
To find the probability of machine D given a defective tablet, we will use Bayes' theorem. Bayes' theorem states that the probability of an event A given event B is equal to the probability of event B given event A, multiplied by the probability of event A, divided by the probability of event B.
Let's denote:
- P(D) as the probability of machine D (0.13)
- P(Def) as the probability of a defective tablet (0.11)
- P(Def|D) as the probability of a defective tablet given that it is produced by machine D (0.05)
Using Bayes' theorem, we can calculate the probability of machine D given a defective tablet:
P(D|Def) = (P(Def|D) * P(D)) / P(Def)
P(D|Def) = (0.05 * 0.13) / 0.11
b) Probability of machine B given a defective tablet:
To find the probability of machine B given a defective tablet, we will use the same approach as in part a.
Let's denote:
- P(B) as the probability of machine B (0.45)
- P(Def|B) as the probability of a defective tablet given that it is produced by machine B (0.2)
Using Bayes' theorem, we can calculate the probability of machine B given a defective tablet:
P(B|Def) = (P(Def|B) * P(B)) / P(Def)
To find the value of P(Def), we will use the law of total probability, which states that the probability of an event is the sum of the probabilities of that event occurring given different conditions.
P(Def) = P(Def|A) * P(A) + P(Def|B) * P(B) + P(Def|C) * P(C) + P(Def|D) * P(D)
P(Def) = (0.11 * 0.23) + (0.2 * 0.45) + (0.07 * 0.19) + (0.05 * 0.13)
Once you calculate P(Def), you can substitute it into the formula for P(B|Def) to find the probability of machine B given a defective tablet.