In a class of 60 students, 25 study French, 30 study German and 10 study both French and German. A student is selected at random from the class. If it is known that the student studies at least one of these two languages, then the probability that the student studies both is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{2}{9}\)
The number of students who study at least one of the two languages is found by the addition rule (the 10 who study both are counted only once):
\(25+30-10 = 45\)
The given condition makes these 45 students the reduced sample space. All 10 students who study both are among them:
\( \displaystyle P = \frac{10}{45} = \frac{2}{9} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{1}{6}\) — Incorrect. This is \(\frac{10}{60}\), which ignores the given condition.
Option (2) \(\displaystyle\frac{1}{3}\) — Incorrect. This is \(\frac{10}{30}\), the probability given that the student studies German.
Option (3) \(\displaystyle\frac{2}{5}\) — Incorrect. This is \(\frac{10}{25}\), the probability given that the student studies French.