Question

In the binomial function, negative binomial function, poisson distribution, I dont know what to do when we need to find a variable X. For example, If X is exactly at 0, 1 , 2, etc. Then I know that we only need to apply the formula and calculate it. However, in some cases like X <= 2, X >= 5, X > 4, etc, then I do not know how to calculate that X and how to apply the formula. Ex: If P( X >= 4) = 1 - P(X <= 3) and for X <= 3, we will calculate the sum of X = 0, X = 1, X = 2, X = 3. How to define when to use 1 - P(X <= 3) or how P(X = 4) = P(X <= 4) - P(X <= 3). It really hard for me to understand this concept. Is there any formula or any way to define it so you know when to subtract, or when to add it together? Thank you.

Answer 1

Solution: In order to explain, let's consider an example of binomial probability with

The binomial probability distribution for is given below:

Now, let's suppose we have to find

So it means we have to find Probability less than or equal to 2.

There are two ways to find it.

1.

  

2. Using the complementary law of probability

  

  

  

So using both the methods we got the same probability, but the difference is second method needs more calculations than first method. So method one is preferred here.

Similarly if we are required to find

We have two methods to solve it:

1.

  

2. Using the complementary law of probability

So using both the methods we got the same probability, but the difference is here that first method needs more calculations than second method. So method second is preferred here.

So the moral of the story is that we can use any of these two methods to get the required probability but the method with less calculations is prefered.

Hope this helps