In the previous example, we only discussed the probability of getting an exact value. For example, we know the probability of X = 4 (4 red balls). But what if the house wants to know the probability of getting < 3 red balls, as the house knows that for < 3 red balls, the players will lose and the house will make money?

Sometimes, talking in terms of **less than** is more useful. For example — how many employees can get to work in less than 40 minutes? Let’s explore how you can find the probability for such cases.

**Clarification:** The cumulative probability distribution table, shown in the above video (02:22 to 03:11) should be a follows:

x | F(x) = P(X<x) |

0 | 0.0256 |

1 | 0.1792 |

2 | 0.5248 |

3 | 0.8704 |

4 | 1.0000 |

(**Side Note**: In the question for calculating the probability of weights between 60 and 65, you might have noticed that we calculated P(X ≤ 65)-P(X ≤ 60) to find P(60 ≤ X ≤ 65).

However, this is not entirely correct if you consider discrete distributions, because P(X=60) would also get subtracted from the value P(X<=65) when we evaluate P(X ≤ 65)-P(X ≤ 60)

The assumption we made, that both P(X=60) and P(X=65) would be equal to 0, is necessary for the calculation to hold true. However, an important concept that you will learn further

is that the “weight” variable is generally considered to be continuous and not discrete, and for continuous variables, the value of P(X=x), where X is a random variable and x is a value, is always 0. Hence, the logic provided in the video holds. So, for cumulative distribution tables where the probabilities of individual values are not given, you can just use a similar analogy to

calculate the probability between the two values.)

So, the **cumulative probability of X**, denoted by **F(x)**, is defined as **the probability of the variable being less than or equal to x**.

In mathematical terms, you would write cumulative probability **F(x) = P(X <x)**. For example, F(4) = P(X<4), F(3) = P(X<3).

For binomial distribution, you can visualise the cumulative probability using this interactive app. (Use the green slider below the probability distribution to see what the cumulative probability looks like.)

This concept of cumulative probability will be used extensively in the next session, where we will talk about continuous random variables.