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?

Short answer

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

Step-by-step solution

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.