Question

In a vaccination drive, there are three types of vaccine $A,B,C$ are available. If $30\%$ population has taken dose of $A$ but not $B,35\%$ has taken dose of $B$ but not $C,20\%$ has taken dose of $C$ but not $A,45\%$ has taken combination of exactly two. Then the percentage of population vaccinated through exactly one type of vaccine is :

• A. $30$
• B. $60$
• C. $50$
• D. $40$

Answer 1

The correct option is D $40$

Given :
$P(A\cap B^c)=\frac{30}{100},P(B\cap C^c)=\frac{35}{100},P(C\cap A^c)=\frac{20}{100},P(A\cap B)+P(B\cap C)+P(C\cap A)-3P(A\cap B\cap C)=\frac{45}{100}$
Now, Required probability
$=P(\text{exactly one of}A,B,C)=P\left(A\right)+P\left(B\right)+P\left(C\right)-2P(A\cap B)-2P(B\cap C)-2P(C\cap A)+3P(A\cap B\cap C)=P(A\cap B^c)+P(B\cap C^c)+P(C\cap A^c)-\left[P\right(A\cap B)+P(B\cap C)+P(C\cap A)-3P(A\cap B\cap C\left)\right]=\frac{30}{100}+\frac{35}{100}+\frac{20}{100}-\frac{45}{100}=40\%$