A and B are participating in the same contest; the probability of their winning are \(\frac{5}{9}\) and \(\frac{1}{4}\) respectively. What is the probability that neither of them will win the contest?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\frac{1}{3}\)
Note: The given answer is as per the official NTA Answer Key. However, the question has two possible answers, i.e. both Option (1) and Option (4), as explained below.
The question does not say whether the contest can have more than one winner, and the answer depends on it.
If the two results are independent (both A and B can win, as in a contest where everyone above a score wins), the probability that neither wins is the product of the probabilities of not winning:
A does not win: \(1-\frac{5}{9}=\frac{4}{9}\)
B does not win: \(1-\frac{1}{4}=\frac{3}{4}\)
Neither wins: \(\displaystyle\frac{4}{9}\times\frac{3}{4}=\frac{1}{3}\), which is Option (4)
If the contest has only one winner, A winning and B winning cannot both happen (they are mutually exclusive), so their probabilities are added:
A or B wins: \(\displaystyle\frac{5}{9}+\frac{1}{4}=\frac{20+9}{36}=\frac{29}{36}\)
Neither wins: \(\displaystyle1-\frac{29}{36}=\frac{7}{36}\), which is Option (1)
Tip for students: When a question has two possible answers, mark the one you worked out yourself. If NTA’s answer key gives a different answer, challenge it when NTA opens the window for challenges.