Earlier, we found the theoretical probability for our game and compared it with the experimental one. Finding the probability without conducting an experiment means that we can find the probability using just pen and paper and with minimal effort.

Now, let’s try to generalise it — let’s say that the probability of getting 1 red ball in one trial is equal to p. In that case, what would be the probability of all 4 balls being red? Let’s see in the following video.

So, the probability distribution for X (i.e., the number of red balls drawn after 4 trials) if the **probability of getting a red ball in 1 trial is ‘p’ is as follows**.

In the following video, we will see how this can be generalised even further.

So, the formula for finding **binomial probability** is given by:

P(X=r)= nCr(p)r(1−p)n−r

Where **n** is **the number of trials**, **p** is the **probability of success**, and **r** is the **number of successes after n trials**.

However, there are some **conditions** that need to be met in order for us to be able to apply the formula.

- The
**total number**of trials is**fixed**at**n**. - Each trial is
**binary**, i.e., it has**only two possible outcomes:**success or failure. **Probability of success**is the**same**in all trials, denoted by**p**.