What is the probability that the dress was made by the first machine?

Question

In a factory there are two machines making clothes. Each machine makes mistakes, for the first one $\lambda = 0,2$ and for the second one $\lambda = 0,5$. Let's suppose that there were two defects on the dress we have bought. What is the probability that the dress was made by the first machine?

Of course I would be happy for a solution, but I don't even understand this exercise. Can someone help me with some explanation what is this exercise is about?

Answer 1

This is a typical Bayes' theorem question: if you knew which machine made it, you would know the probability of two defects, and now you know that there were two defects and you want to invert to try to guess which machine made it. Bayes' theorem is usually stated as

$$P(A \mid B)= \frac{P(B \mid A)P(A)}{P(B \mid A) P(A) + P(B \mid A^c) P(A^c)}.$$

So take $A$ to be the event that the first machine made the dress and $B$ to be the event of there being two defects.

The ingredients you need now are $P(B \mid A),P(B \mid A^c)$ and $P(A)$ (and $P(A^c)$ but that is trivially determined from $P(A)$). The first two numbers are known from the Poisson distribution. The last number "really" has a prior distribution which can't "really" be determined out of context (since nothing was given about it). However, the "naive prior" has $P(A)=1/2$ (you assume the machine was chosen uniformly at random). That is probably what the author of the question expects you to use.

Answer 2

There is a tacit assumption that the dress you bought is initially equally likely to have come from either machine. That would makes sense if we assume they both produce garments at the same rate.

Presumably the values of $\lambda$ are in errors per unit of time: it might be errors per hour or errors per day or errors per month, etc. For the present problem we don't need to know how much time it is.

In English-speaking countries, as far as I know, one normally writes $0.2$ rather than $0{,}2$ when what is meant is $2/10.$

If one assumes equally many dresses are made by both machines then on average, of $10$ units of time (hours, days, months, or whatever they are) one machine makes $2$ mistakes and the other $5,$ for a total of $7.$ If you get one that has such a defect, then the probability is $2/7$ that it came from the first machine and $5/7$ that it is from the second.