Question

Suppose three test kits are tested at random. Let N be the random variable for the number of non - defective test kits, construct the probability distribution P(N). What can you conclude about the effectivity of the COVID-19 Rapid Antibody Test kits based on your probability distribution? Explain briefly.

Answer 1

Solution:

Let X₁, X₂ and X₃ are three kits that are tested random

$X_{i,\ i=1,2,3}=1\ if\ X_i=D \ \ \ \ \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =0\ if \ X_i=N$

Let, total number of test kits that are defective =d 

and total number of test kits that are non-defective =n

$\therefore P[x_ i=D]=\frac{d}{d+n}=p(\operatorname{let})$

Then, $x=\sum_{i=1}^{3} x i=$  number of defective test kits.

Now, $x_{i} \sim$  bernoulli (p)

$\therefore \sum_{i=1}^{3} x i \sim$  binomial (3, p) where, $p=\frac{d}{d+n}$

$\begin{aligned} \therefore P[X=x] &=\left(\begin{array}{l} 3 \\ x \end{array}\right) p^{x}(1-p)^{3-x}, x=0,1,2,3 \\ &=0 \text { otherwise. } \end{aligned}$