Question

Two 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 from the two companies. Only 80% of the Y company cellphones appeared to be correctly identified, and 20% of the Y company cellphones were misclassified asa productfrom, 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

$Let~C~be~event~that~the~witness~identified~the~cellphone~correctly\\ Let~A~be~an~event~that~the~cellphone~really~belong~to~Y~company.\\ Hence~P(C|A)=0.80,~P(A)=0.3.\\ Required~probability=Probability~that~the~witness~identified~the~cellphone\\ correctly~and~the~cellphone~really~belonged~to~Y~company=P(A\cap C)\\ =P(C|A)P(A)=0.8*0.3=0.24$