Vaccinated and ill

Problem

A population of $10\,000$ individuals is $60\%$ vaccinated. Among the vaccinated, $10\%$ fall ill; among the unvaccinated, $30\%$. Fill in the contingency table. How many vaccinated individuals fall ill? What is the probability that an ill person chosen at random was not vaccinated?

Solution

Vaccinated: $10\,000\cdot 0,6 = 6000$; unvaccinated: $4000$.

Contingency table:

	ill	healthy	total
vaccinated ($V$)	$600$	$5400$	$6000$
unvaccinated ($\overline{V}$)	$1200$	$2800$	$4000$
total	$1800$	$8200$	$10\,000$

The vaccinated who fall ill are $6000\cdot 0,1 = \boxed{600}$; the unvaccinated who are ill are $4000\cdot 0,3 = 1200$, for a total of $1800$ ill. The probability that an ill person was not vaccinated is $P(\overline{V}\mid \text{ammalato}) = \frac{1200}{1800} = \boxed{\dfrac{2}{3}}$

Links

Topics: Probabilita
Concepts: Probabilita condizionata · Tabella di contingenza
Methods: Tabella contingenza
Skills: Calcolo probabilita
Exercise type: Problema probabilita