91Ó°ÊÓ

Students in Spanish class$=P\left(A\right)=0.28$

Students in French class$=P\left(B\right)=0.26$

Students in German class$=P\left(C\right)=0.16$

Students who are in both Spanish and French class role="math" localid="1649346042059" $=P(Aâˆ\copyright B)=0.12$

Students who are in both Spanish and German class $=P(Aâˆ\copyright C)=0.04$

Students who are in both French and German class $=P(Bâˆ\copyright C)=0.06$

Students taking all three classes $=P(Aâˆ\copyright Bâˆ\copyright C)=0.02$

The probability that the chosen student is not in any of the language classes $P{\left(A∪B∪C\right)}^c$.

role="math" localid="1649346573978" $P{\left(A∪B∪C\right)}^c=1-P\left(A∪B∪C\right)$

role="math" localid="1649346486028" $$\begin{gathered} P\left(A∪B∪C\right)=P\left(A\right)+P\left(B\right)+P\left(C\right)-P\left(Aâˆ\copyright B\right)-P\left(Bâˆ\copyright C\right)-P\left(Aâˆ\copyright C\right)+P{\left(A∪B∪C\right)}^c \\ =0.28+0.26+0.16-0.12-0.04-0.06+0.02 \\ =0.5 \end{gathered}$$

$$\begin{gathered} P{\left(A∪B∪C\right)}^c=1-0.5 \\ =0.5 \end{gathered}$$

Therefore, the probability that the chosen student is not in any of the language classes is$0.5$.

The probability that the chosen student is taking exactly one language class $=P\left(Aâˆ\copyright B^câˆ\copyright C^c\right)+P\left(A^câˆ\copyright Bâˆ\copyright C^c\right)+P\left(A^câˆ\copyright B^câˆ\copyright C\right)$

$$\begin{gathered} P\left(Aâˆ\copyright B^câˆ\copyright C^c\right)=P\left(A\right)-\left[P\left(Aâˆ\copyright B\right)+P\left(Aâˆ\copyright C\right)-P\left(Aâˆ\copyright Bâˆ\copyright C\right)\right] \\ =0.28-\left[0.12+0.04-0.02\right] \\ =0.14 \\ P\left(A^câˆ\copyright Bâˆ\copyright C^c\right)=P\left(B\right)-\left[P\left(Aâˆ\copyright B\right)+P\left(Bâˆ\copyright C\right)-P\left(Aâˆ\copyright Bâˆ\copyright C\right)\right] \\ =0.26-\left[0.12+0.06-0.02\right] \\ =0.10 \\ P\left(A^câˆ\copyright B^câˆ\copyright C\right)=P\left(C\right)-\left[P\left(Aâˆ\copyright C\right)+P\left(Bâˆ\copyright C\right)-P\left(Aâˆ\copyright Bâˆ\copyright C\right)\right] \\ =0.16-\left[0.04+0.06-0.02\right] \\ =0.08 \\ P\left(Aâˆ\copyright B^câˆ\copyright C^c\right)+P\left(A^câˆ\copyright Bâˆ\copyright C^c\right)+P\left(A^câˆ\copyright B^câˆ\copyright C\right)=0.32 \end{gathered}$$

Therefore, the probability that the chosen student is taking exactly one language class isrole="math" localid="1649347106423" $0.32$.

The probability that the chosen student is not in any of the language classes is $0.5$.

Total no. of students in class $=100$.

No. of students not in any of the language classes $=100×0.5=50$

Probability that the two students chosen randomly are not taking any random class is

$$\begin{gathered} =\frac{\left(\begin{array}{c}50\\ 2\end{array}\right)}{\left(\begin{array}{c}100\\ 2\end{array}\right)} \\ =\frac{50×49}{100×99} \\ =\frac{49}{198} \end{gathered}$$

Therefore, the probability that at least one of the chosen student is taking a language class is

$$\begin{gathered} =1-\frac{49}{198} \\ =0.7525 \end{gathered}$$