10% of the population in a town is HIV+ . A new diagnostic kit for HIV detection is available; this kit correctly identifies HIV+ individuals 95% of the time, and HIV− individuals 89% of the time. A particular patient is tested using this kit and is found to be positive. The probability that the individual is actually positive is _______
Concept:
Bayes' Theorem:
Let E1, E2, ….., En be n mutually exclusive and exhaustive events associated with a random experiment and let S be the sample space. Let A be any event which occurs together with any one of E1 or E2 or … or En such that P(A) ≠ 0. Then
\(P\left( {{E_i}\;|\;A} \right) = \frac{{P\left( {{E_i}} \right)\; \times \;P\left( {A\;|\;{E_i}} \right)}}{{\mathop \sum \nolimits_{i\; = 1}^n P\left( {{E_i}} \right) \times P\left( {A\;|\;{E_i}} \right)}},\;i = 1,\;2,\; \ldots .\;,\;n\)
Calculation:
The patient is actually HIV +ve.
So probability of that = P(he belongs to HIV +ve population) * P( HIV +ve test gives correct result given he is HIV +ve).
∴ Probabilty P(A) = 0.1 and P(E/A) = 0.95
The next case is a patient is HIV -ve but the test is showing HIV +ve i.e. test shows wrong result.
So, P(B) = 0.9 and P(E/B) = 1 - 0.89 = 0.11
So By Bayes' Theorem,
P(A/E) i.e. P(actually is HIV +ve) = \(\frac{{\left\{ {{\rm{P}}\left( {\rm{A}} \right){\rm{\times\;P}}\left( {\frac{{\rm{E}}}{{\rm{A}}}} \right)} \right\}}}{{\left[ {{\rm{\;}}\left( {{\rm{P}}\left( {\rm{A}} \right){\rm{\;\times\;P}}\left( {{\rm{E}}/{\rm{A}}} \right)} \right){\rm{\;}} + {\rm{\;}}\left( {{\rm{P}}\left( {\rm{B}} \right){\rm{\;\times\;P}}\left( {{\rm{E}}/{\rm{B}}} \right)} \right){\rm{\;}}} \right]}}{\rm{\;}}\)
\( = \frac{{0.1\;\times\;0.95}}{{\;\left[ {\left( {0.1\;\times\;0.95} \right)\; + \;\left( {0.9\;\times\;0.11} \right)} \right]}}\)
\( = \frac{{0.095}}{{\;0.095\; + \;0.099}}\)
= 0.095 / 0.194
= 95 / 194
= 0.4897
= 48.97 %
Important Questions from Conditional Probability
Let A and B be two events such that \(P(\overline{A \cup B}) = \dfrac{1}{6}\) , P(A ∩ B) = \(\dfrac{1}{4}\) and P(A̅) = \(\dfrac{1}{4}\) , where A̅ stands for complement of event A. Then, events A and B are:
A bike manufacturing factory has two plants P and Q. Plant P manufactures 60 percent of bikes and plant Q manufacture 40 percent. 80 percent of the bikes at plant P and 90 percent of the bikes at plant Q are rated of standard quality. A bike is chosen at random and is found to be of standard quality. What is the probability that it has come from plant P?
A and B are two events such that A̅ and B̅ are mutually exclusive. If P(A) = 0.5 and P(B) = 0.6, then what is the value of P(A|B)?
For two dependent events A and B, it is given that P(A) = 0.2 and P(B) = 0.5. If A ⊆ B, then the values of conditional probabilities P(A|B) and P(B|A) are respectively
In a bulb factory, machines P, Q and R manufacture respectively 25%, 35% and 40% of the total. Of their output 5, 4 and 2 percent respectively are defective bulbs. A bulb is drawn at random and it is found to be defective. What is the probability that it was manufactured by machine Q?
