# Cumulative binomial distribution tables

This section is a continuation from the previous binomial distribution article. It may be a good idea to go through it before attempting this article. This article explores using tables of the cumulative distribution function of the binomial distribution to find probabilities. The table has been attached to this article.
Using tables of the cumulative distribution function of the binomial distribution to find probabilities can make potentially length calculations much shorter. For the binomial distribution X ~ B(n , p) there are tables giving P(X ≤ x) for various values of n and p.
Below are some examples using tables of binomial cumulative distribution, note that the distribution is given in the form X ~ (n, p) and the probability is given in the form P(X ≤ x). So we will need the n, p and x value; where n is the number of trials, p is the probability of success of each trial.

We shall explore some examples below;

### Example

In this example suppose the random variable X ~ B(20 , 0.4). Find;- P(X ≤ 7)
- P(X < 6)
- P(X ≥ 15)

Note that the tables can also be used to find P(X ≤ x) since [IMAGE]

### Example

Here is another example. Suppose the random variable X ~ B(25, 0.5). Find;- P(X ≤ 6)
- P(X = 6)
- P(X > 13)
- P(6 <x ≤ 10)

## Applications

Now we shall try some applications. Applications that use cumulative probabilities require careful interpretation. There are many different forms of keywords that can be used to reference probabilities. The correct interpretation of these phrases or keywords is critical, especially when dealing with discrete distributions such as; the binomial and Poisson distributions. The following tables explore some of these phrases that you may encounter. P(X ≤ 3)1 – P(X ≤ 7)P(X ≤ 9)P(X ≤ 8)Phrase | Means | Uses |
---|---|---|

…greater than 5… | X > 5 | 1 – P(X ≤ 5) |

…no more than 3… | X ≤ 3 | |

…at least 7… | X ≥ 7 | |

…fewer than 10… | X < 10 | |

…at most 8… | X ≤ 8 |

### Example

Here is an example; A spinner is designed so that the probability it lands on red is 0.3. Jane has 12 spins. Fid the probability that Jane obtains;- no more than 2 reds
- at least 5 reds

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