For the FIFA world cup, Paul the octopus has been predicting the winner of…
2025
For the FIFA world cup, Paul the octopus has been predicting the winner of each match with amazing success. It is rumoured that in a match between 2 teams A and B, Paul picks A with the same probability as A chances of winning. Lets assume such rumours to be true and that in a match between Ghana and Bolivia; Ghana the stronger team has a probability of 2/3 of winning the game. What is the probability that Paul will correctly pick the winner of the Ghana-Bolivia game?
- A.
1/9
- B.
4/9
- C.
5/9
- D.
2/3
Attempted by 1 students.
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Correct answer: C
Concept: By the law of total probability, when an event can occur through several mutually exclusive routes, its overall probability is the sum of the probability of each route. Here, Paul's prediction turns out correct through exactly two mutually exclusive routes: he picks the winning team, for either Ghana or Bolivia.
Let p = P(Ghana wins) = 2/3, so P(Bolivia wins) = 1 − p = 1/3.
Paul's picking rule mirrors the true win probabilities, so P(Paul picks Ghana) = 2/3 and P(Paul picks Bolivia) = 1/3.
Route 1 (correct via Ghana): Paul picks Ghana AND Ghana wins → (2/3) × (2/3) = 4/9.
Route 2 (correct via Bolivia): Paul picks Bolivia AND Bolivia wins → (1/3) × (1/3) = 1/9.
These two routes are mutually exclusive, so add them: P(correct) = 4/9 + 1/9 = 5/9.
Cross-check: in general P(correct) = p2 + (1 − p)2 for win probability p, which is always ≥ 1/2 (minimum at p = 1/2). With p = 2/3, this gives 5/9 ≈ 0.556, consistent with being slightly better than a coin flip — a sensible result since one team is more predictable than the other.
Therefore, the probability that Paul correctly predicts the winner is 5/9.