Overweight person and sport (Bayes)

Text

In a population, $30\%$ of people play sport. Among those who play sport, $5\%$ are overweight; among those who do not, $25\%$. What is the probability that a randomly chosen overweight person plays sport?

Solution

Let $S$ denote the event «plays sport». The data are $P(S)=0,3,\qquad P(\text{sovr}\mid S)=0,05,\qquad P(\text{sovr}\mid\overline{S})=0,25.$ Using the total probability formula we compute the probability of being overweight: $P(\text{sovr}) = 0,3\cdot 0,05 + 0,7\cdot 0,25 = 0,015 + 0,175 = 0,205.$ We then apply Bayes’ theorem: $P(S\mid\text{sovr}) = \frac{P(\text{sovr}\mid S)\,P(S)}{P(\text{sovr})} = \frac{0,015}{0,205} \approx \boxed{7,3\%}$

Links

Topics: Probabilita
Concepts: Probabilita totale · Teorema di bayes
Methods: Bayes · Prob totale
Skills: Calcolo probabilita
Exercise type: Problema probabilita