[Math] probability in statististics

probability

A factory has a machine shop in which three machines (A,B and C) produce 100cm aluminum tubes. An inspector is equally likely to sample tubes from A and B and thrice as likely to select tubes from C as he is from B. The defective rates from the three machines are A=10%,B=10%,C=20%. What is the probability that a tube selected by the inspector. 1.is from machine A? 2. is defective. 3. comes from machine A,given that it is defective.

Best Answer

To get you started, with $(1)$

$P(A) = P(B)$

$P(C) = 3P(B)$

$$P(A) + P(B) + P(C) = 1 \implies 5P(B) = 1 \iff P(B) = 0.2$$

So $P(A) = P(B) = 0.2$, whereas $P(C) = 3P(B) = 0.6$.

$(2)$: The probability that the tube is defective is equal to the sum of the following probabilities:

  • The tube selected was produced by $A$ AND is defective: $\;P(A) \times 0.1$.
  • The tube selected was produced by $B$ AND is defective: $\;P(B) \times 0.1$.
  • The tube selected was produced by $C$ AND is defective: $\;P(C) \times 0.2$.
Related Question