Multiple choice

While watching a game of Champions League football in a cafe, you observe someone who is clearly supporting Manchester United in the game. What is the probability that they were actually born within 25 miles of Manchester? Assume that: $\implies $the probability that a randomly selected person in a typical local bar environment is born within $25$ miles of Manchester is $\dfrac{1}{20}$, and; $\implies$the chance that a person born within $25$ miles of Manchester actually supports United is $\dfrac{7}{10}$; $\implies$the probability that a person not born within $25$ miles of Manchester supports United with probability $\dfrac{1}{10}$

  1. $\dfrac{7}{26}$
  2. $\dfrac{8}{26}$
  3. $\dfrac{9}{26}$
  4. $\dfrac{10}{26}$
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A Correct answer
Explanation

Use Bayes' Theorem. Let M be born in Manchester, S be support United. P(M|S) = [P(S|M) * P(M)] / [P(S|M) * P(M) + P(S|not M) * P(not M)]. P(M|S) = [(7/10)(1/20)] / [(7/10)(1/20) + (1/10)*(19/20)] = 7 / (7 + 19) = 7/26.

AI explanation

Using Bayes' theorem, let A be the event of being born within 25 miles of Manchester and B be the event of supporting United. We calculate P(A given B) as the probability of B given A multiplied by P(A), divided by the total probability of B. Substituting the values gives (7/10 times 1/20) divided by ((7/10 times 1/20) + (1/10 times 19/20)), which equals 7/200 divided by 26/200, resulting in 7/26.

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