Question

T'wo cellphone companies operate in a given city, the X company and the Y company. The X company is more popular than the Y company, as you know that 70% of the users belong to the X company and the remaining 30% belong to the Y company. A cell call was involved in a hit and run accident at night. A witness identified the cellphone involved had the product icon from the Y company. The court tested the appearance of products from the two companies under similar visibility conditions. When testing a sample of cellphones, the court found some identification errors can occur due to similar design of the cellphone products form the two companies. Only 80% of the Y company cellphones appeared to be correctly identified, and 20% of the Y company cellphones were misclassified as a product from the X company. Only 75% of the X company cellphones appeared to be correctly identified, and 25% of the X company cellphones were misclassified as a product from the Y company. Hence, even though the witness identified the cellphone as a produce from the Y company, it could have belonged to the X company. What is the probability that the witness identified the cellphone correctly and the cellphone really belonged to the Y company? Make sure that you show your working.

Answer 1

C=event of identifying cell phone correctly,

X=event that cell phone belongs to the X company

Y=event that cell phone belongs to the Y company

According tho the problem, P(C|Y)=0.80, P(C|X)=0.75, P(X)=0.70, P(Y)=0.30

Probability that the witness identified the cellphone correctly and the cellphone really belong to the Y company=P(Y|C)

=P(C|Y)P(Y)/(P(C|X)P(X)+P(C|Y)P(Y))=0.80*0.30/(0.75*0.70+0.80*0.30)=0.3137