Question

While watching a game of champions league football in a cafe, $\text{Andrew}$ observe someone who is clearly supporting Manchester United in the game. The probability that they were actually born within $25$ miles of Manchester.
Assume that:
$\cdot$ The probability that a randomly selected person in a typical local bar environment is born within $25$ miles of Manchester is $\frac{1}{20}.$
$\cdot$ The chance that a person born within $25$ miles of manchester actually supports united is $\frac{7}{10}.$
$\cdot$ The probability that a person not born within $25$ miles of Manchester supports United with probability $\frac{1}{10}.$

• A. $\frac{1}{26}$
• B. $\frac{7}{26}$
• C. $\frac{25}{26}$
• D. $\frac{3}{26}$

Answer 1

The correct option is B $\frac{7}{26}$

Let, $B$ be an event that the person is born within $25$ miles of Manchester, and $U$ be an event that the person supports United.
By Bayes' Theorem
$P\left(B|U\right)=\frac{P\left(U|B\right)P\left(B\right)}{P\left(U|B\right)P\left(B\right)+P\left(U|B^{\prime }\right)P\left(B^{\prime }\right)}$
$=\frac{\frac{7}{10}\cdot \frac{1}{20}}{\frac{7}{10}\cdot \frac{1}{20}+\frac{1}{10}\cdot \frac{19}{20}}=\frac{7}{26}$