Q.4.8 - A First Course in Probability

Question

Let $B(n, p)$ represent a binomial random variable with parameters $n$ and $p$. Argue that

 $P\left\{B\right(n,p)\le i\}=1-P\left\{B\right(n,1-p)\le n-i-1\}$

 Hint: The number of successes less than or equal to $i$ is equivalent to what statement about the number of failures? 

Step-by-Step Solution

Answer

Events $B(n,p)\le i$ and $B(n,1-p)\ge n-i$ are equivalent.

Step 1 : Given information

Let $B(n, p)$ represent a binomial random variable with parameters $n$ and $p$. Argue that

$P\left\{B\right(n,p)\le i\}=1-P\left\{B\right(n,1-p)\le n-i-1\}$

Step 2 : Explanation

Assume that$B(n, p)$ is counter of successes and $B(n, 1-p)$ is simply $n-B(n, p)$. Observe that the required equality can be written as

$P\left(B\right(n,p)\le i)=P\left(B\right(n,1-p)>n-i-1)=P\left(B\right(n,1-p)\ge n-i)$

It is enough to show that events $B(n,p)\le i$ and $B(n,1-p)\ge n-i$ are equivalent.

The event $B(n,p)\le i$ means that we have obtained less or equal to $i$ successes. So, that is equivalent to the information that we have obtained more or equal to $n-i$ failures, which is the second event. Hence, we have proved the claimed.

Step 3 : Final answer

Events $B(n,p)\le i$and $B(n,1-p)\ge n-i$ are equivalent.