Suppose the probability of getting a job interview from a single application is $p$, and you submit $n$ independent applications. Which expression gives the probability of getting at least one interview if you submit between $1$ and $10$ applications?

• A. $p^n$
• B. $1-p$
• C. $n\ast p$
• D. $1-{(1-p)}^n$

Answer: (D)

Verified step by step guidance

1. Understand the problem: You want to find the probability of getting at least one job interview after submitting \( n \) independent applications, where each application has a probability \( p \) of resulting in an interview.
2. Recall the complement rule in probability: The probability of at least one success is equal to 1 minus the probability of no successes. Here, "no successes" means getting zero interviews from all \( n \) applications.
3. Calculate the probability of no interviews: Since each application is independent and the probability of an interview is \( p \), the probability of not getting an interview from a single application is \( 1 - p \). For \( n \) independent applications, the probability of no interviews is \( (1 - p)^n \).
4. Apply the complement rule: The probability of getting at least one interview is therefore \( 1 - (1 - p)^n \).
5. Interpret the expression: This formula accounts for all cases where you get one or more interviews out of the \( n \) applications, which matches the problem's requirement.