A laboratory blood test is 99 % effective in detecting a certain … - Vidyadip

Question

A laboratory blood test is $99 \%$ effective in detecting a certain disease when its infection is present. However, the test also yields a false positive result for $0.5 \%$ of the healthy person tested (i.e. if a healthy person is tested, then, with probability $0.005$ , the test will imply he has the disease). If $0.1 \%$ of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?

✓

Answer

Let $E _1$ and $E _2$ denote the events that a person has a disease and a person has no disease, respectively. $E _1$ and $E _2$ are complimentary to each other.
$\therefore$ $P(E_1) + P(E_2) = 1$
$\Rightarrow P(E_2) = 1 - P(E_1) = 1 - 0.001 = 0999$
let A denote the event that the blood test result is positive.
$\therefore$ $P(E_1) = 0.1% = 0.001$
Now,
$\text{P}\Big(\frac{\text{A}}{\text{E}_1}\Big)=90\%=0.99$
$\text{P}\Big(\frac{\text{A}}{\text{E}_2}\big)=0.5\%=0.005$
Using Baye's theorem, we get
Required probability $\text{P}\Big(\frac{\text{E}_1}{\text{A}}\Big)=\frac{\text{P}(\text{E}_1)\text{P}\Big(\frac{\text{A}}{\text{E}_1}\Big)}{\text{P}(\text{E}_1)\text{P}\Big(\frac{\text{A}}{\text{E}_1}\Big)+\text{P}(\text{E}_2)\text{P}\Big(\frac{\text{A}}{\text{E}_2}\Big)}$
$=\frac{0.001\times0.99}{0.001\times0.99+0.999\times0.005}$
$=\frac{990}{5985}=\frac{22}{133}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Probability→Probability — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following integrals:
$\int\frac{1}{\sqrt{\tan^{-1}\text{x}}.(1+\text{x}^2)}\text{dx}$

Solve: $2 y e ^{ x / y } d x+\left(y-2 x e ^{ x / y }\right) dy =0$

A box contains 13 bulbs, out of which 5 are defective. 3 bulbs are randomly drawn, one by one without replacement, from the box. Find the probability distribution of the number of defective bulbs.

If $\text{A}=\begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix},$ then show that $\text{A}^2=\begin{bmatrix}\cos2\theta&\sin2\theta\\-\sin2\theta&\cos2\theta\end{bmatrix}.$

Using Rolle’s theorem, find the point on the curve $\text{y}=\text{x}(\text{x}-4),\text{x}\in[0,4].$ where the tangent is parallel to x-axis.

Prove that in set $Z$ relation $R$ defined as $a R b$ $\Leftrightarrow a-b$ is divisible by 3 , is equivalence relation.

Evaluate: $\int \frac{x^2+1}{\left(x^2+2\right)\left(2 x^2+1\right)} d x$

Let $*$ be the binary operation defined on $Q$. Find which of the following binary operations are commutative:

$a * b = a – b \forall a, b \in Q$
$a * b = a^2 + b^2 \forall a, b \in Q$
$a * b = a + ab \forall a, b \in Q$
$a * b = (a – b)^2 \forall a, b \in Q$

Evaluate the following integrals:
$\int\frac{1}{\text{x}}(\log\text{x})^2\text{dx}$

Write the value of the determinant $\begin{vmatrix}\text{x}+\text{y}&\text{y}+\text{z}&\text{z}+\text{x}\\\text{z}&\text{x}&\text{y}\\-3&-3&-3 \end{vmatrix}$