The probability that a person with certain symptoms has hepatitis is 0.75. The blood test used to confirm this diagnosis gives positive results for 86?% of people with the disease and 5?% of those without the disease. What is the probability that an individual who has the symptoms and who reacts positively to the test actually has? hepatitis?

Answer:0.981Step-by-step explanation:We have to find the probability that an individual who has the symptoms and who reacts positively to the test has hepatitisLet [tex]E_1[/tex]be the event that denotes the symptoms has hepatitis and [tex]E_2[/tex] denotes the symptoms have not hepatitis Let A be the event the denotes the blood test result positive  We have to find the value of [tex]P(E_1/A)[/tex]We have [tex]P(E_1)=0.75[/tex][tex]P(E_2)=1-0.75=0.25[/tex][tex]P(A/E_1)=86%=0.86[/tex][tex]P(A/E_2)=5%=0.05[/tex]Using formula [tex]P(E_1/A)=\frac{P(E_1)\times P(A/E_1)}{P(E_1)\cdotP(A/E_1)+P(E_2)\cdotP(A/E_2}[/tex]Substitute the values in the given formulaThen,we get [tex]P(E_1/A)=\frac{0.75\times .86}{0.75\times0.86+0.25\times 0.05}[/tex][tex]P(E_1/A)=\frac{0.645}{0.6575}=0.9809[/tex][tex]P(E_1/A)=0.981[/tex]Hence, the probability that an individual who has the symptoms and who reacts positively to the test actually has hepatitis =0.981