Answer to Question #219256 in Statistics and Probability for Enam

Question

An item produced by a company is susceptible to two types of defects namely, A and B. The probability that an item has defect A is 1/6 . The probability that it has defect B is 1/8 . Calculate the probability that the item has

(i) Both A and B defects

(ii) Only defect A

(iii) Only defect B

(iv) One defect only

(v) At least one defect

Answer 1

(i) Since the event $A$ and event $B$ are independent, then

$P(A\cap B)=P(A)P(B)=\dfrac{1}{6}(\dfrac{1}{8})=\dfrac{1}{48}$

(ii)

$P(only\ A)=P(A\cap B')=P(A)-P(A\cap B)$

$=\dfrac{1}{6}-\dfrac{1}{48}=\dfrac{7}{48}$

(iii)

$P(only\ B)=P(B\cap A')=P(B)-P(A\cap B)$

$=\dfrac{1}{8}-\dfrac{1}{48}=\dfrac{5}{48}$

(iv)

$P(only\ one)=P(A\cap B')+P(B\cap A')$

$=\dfrac{7}{48}+\dfrac{5}{48}=\dfrac{1}{4}$

(v)

$P(at\ least\ one)=P(A\cup B)$

$=P(A)+P(B)-P(A\cap B)$

$=\dfrac{1}{8}+\dfrac{1}{6}-\dfrac{1}{48}=\dfrac{13}{48}$