At the peak of flu season, one in a hundred gets the flu. But half of the infected show no symptoms. People with allergies or colds can also show Flu symptoms; one in twenty people who don’t have flu can show flu-like symptoms. So the question is: if a person shows signs of the flu, what is the probability that she has the flu?
We will use this example to illustrate problem-solving through Bayes’ rule. So what is the ask here? You have to tell the chance that a person has flu, given she is showing symptoms, S. Or in shorthand, P(F|S). But what do all we know about flu?
- If a random person is picked from the street, there is a one in a hundred chance that he has flu. In other words, P(F) = 1/100 = 0.01. It also means that 99 out of 100 random people in the street have no flu, P(nF) = 0.99.
- Only half of the people who have flu show any symptoms. The probability of expressing signs given the person has flu = 0.5. In shorthand, it is P(S|F).
- 1 in 20 people who don’t have flu can show flu-like symptoms or P(S|nF) = 0.2
Use all the above information and plug it into Bayes’ equation.
P(F|S) = P(S|F) x P(F) /[P(S|F) x P(F) + P(S|nF) x P(nF)] = 0.5 x 0.01 /[0.5 x 0.01 + 0.2 x 0.99] = 0.0246 = 2.5%
