Question

A candidate is selected for interview for three posts. For the first post there are 5 candidates, for the second there are 8 and for the third there are 7. What are the chances for his getting at least one post?

• A. \(\dfrac{1}{5}\)
• B. \(\dfrac{3}{5}\)
• C. \(\dfrac{2}{5}\)
• D. \(\dfrac{4}{5}\)

Hint:

For independent opportunities, "at least one" is fastest via complements: \(1 - \prod (1 - p_i)\). It avoids messy inclusion-exclusion with many terms.

Answer 1

The Correct Option is C

Step 1: Model each post as an independent fair choice among candidates. 
Probability he gets the first post \(= \dfrac{1}{5}\); second \(= \dfrac{1}{8}\); third \(= \dfrac{1}{7}\). 
We seek \(P(\text{at least one post})\). 

Step 2: Use the complement (none of the posts). 
\(P(\text{none}) = \left(1 - \dfrac{1}{5}\right)\left(1 - \dfrac{1}{8}\right)\left(1 - \dfrac{1}{7}\right) = \dfrac{4}{5}\cdot\dfrac{7}{8}\cdot\dfrac{6}{7}\). 
Cancel \(7\): \(= \dfrac{4}{5}\cdot\dfrac{6}{8} = \dfrac{4}{5}\cdot\dfrac{3}{4} = \dfrac{3}{5}\). 

Step 3: Convert back to "at least one". 
\(P(\text{at least one}) = 1 - P(\text{none}) = 1 - \dfrac{3}{5} = \boxed{\dfrac{2}{5}}\).