A hockey match is played between team Blue and team Red. Each team has 8 players, and 4 players of each team are chosen at random to play. Rhea is in team Blue and Vaishali is in team Red. What is the probability that at least one of Rhea and Vaishali is among the players chosen to play?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\frac{3}{4}\)
Each team chooses its players separately, so the two events are independent. “At least one is chosen” fails only when neither is chosen, so its probability is 1 − P(neither).
P(Rhea chosen) \(\displaystyle =\frac{4}{8}=\frac{1}{2}\), so P(Rhea not chosen) \(\displaystyle =\frac{1}{2}\)
P(Vaishali chosen) \(\displaystyle =\frac{4}{8}=\frac{1}{2}\), so P(Vaishali not chosen) \(\displaystyle =\frac{1}{2}\)
P(neither chosen) \(\displaystyle =\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}\)
P(at least one chosen) \(\displaystyle =1-\frac{1}{4}=\frac{3}{4}\)
Why the other options are incorrect
Option (1) \(\frac{3}{5}\) — Incorrect. The chance that at least one is chosen is \(1-\frac{1}{4}=\frac{3}{4}=0.75\), which is more than \(\frac{3}{5}=0.6\).
Option (3) \(\frac{1}{4}\) — Incorrect. This is the probability that neither is chosen (or that both are chosen).
Option (4) \(\frac{2}{5}\) — Incorrect. This is less than the chance of Rhea alone being chosen (\(\frac{1}{2}\)), so it is too small for “at least one”.