Two candidates X and Y contest an election. The probability that X wins is 0.4 and the probability that Y wins is 0.6. If X wins, the probability that a new bridge is built in the town is 0.8; if Y wins, this probability is 0.3. It is later found that a new bridge has been built. The probability that Y won the election is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{9}{25}\)
Let \(E_1\) and \(E_2\) be the events that X wins and Y wins, and let B be the event that the bridge is built.
\(P(E_1) = 0.4,\quad P(B|E_1) = 0.8\)
\(P(E_2) = 0.6,\quad P(B|E_2) = 0.3\)
By the theorem of total probability:
\(P(B) = 0.4\times 0.8+0.6\times 0.3 = 0.32+0.18 = 0.5\)
By Bayes’ theorem:
\( \displaystyle P(E_2|B) = \frac{P(E_2)P(B|E_2)}{P(B)} = \frac{0.18}{0.5} = \frac{18}{50} = \frac{9}{25} \)
Why the other options are incorrect
Option (2) \(\displaystyle\frac{16}{25}\) — Incorrect. This is \(\frac{0.32}{0.5}\), the chance that X won.
Option (3) \(\displaystyle\frac{3}{10}\) — Incorrect. This is P(B | Y wins), the condition taken the wrong way round.
Option (4) \(\displaystyle\frac{1}{2}\) — Incorrect. This is P(B), the overall chance that the bridge is built.