The probability that a hospital patient has a particular disease is $p$. A test for the disease ha

Question

The probability that a hospital patient has a particular disease is $p$. A test for the disease has a probability of $0.99$ of giving a positive result when the patient has the disease and a probability of $0.95$of giving a negative result when the patient does not have the disease. A patient is given the test. For a general value of $p$, the probability that a randomly chosen patient has the disease given that the result of the test is positive is denoted by $\mathrm{f}\left( p \right)$. Find an expression for $\mathrm{f}\left( p \right)$ and show that $\mathrm{f}$ is an increasing function for $0 < p < 1$. Explain what this statement means in the context of this question.[5] For events $A$ and $B$, it is given that $\mathrm{P}\left( A \right)=0.7$ and $\mathrm{P}\left( B \right)=k$. Given that $A$ and $B$ are independent events, find $\mathrm{P}\left( A\cap {B}' \right)$ in terms of $k$. [1] Given instead that $A$ and $B$ are mutually exclusive events, state the range of values of $k$. Find $\mathrm{P}\left( B\mid {A}' \right)$ in terms of $k$.[2]

Accepted Answer

(a) $f(p)=\dfrac{99p}{94p+5}$, $f'(p)=\dfrac{495}{(94p+5)^2}>0$. A positive test is more likely to indicate disease when prevalence increases (b)(i) $\dfrac{7(1-k)}{10}$ (ii) $0\leq k\leq0.3$, $\mathrm P(B\mid A')=\dfrac{10k}{3}$

2019 RI P2 Q5

2019 RI P2 Q5

Video Solution:

Solution:

Solution locked