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.
In this example suppose the random variable X ~ B(20 , 0.4). Find;
- P(X ≤ 7)
- P(X < 6)
- P(X ≥ 15)
First we shall start with P(X ≤ 7), we use the tables of cumulative distribution to find where n=20, p=0,4 and x=7. We find that;
Next we shall find P(X < 6). It is always a good idea to note down the values that you have. We know that;
So we’re finding P(X ≤ 5), we have n=20, p=0.4 and x=5. Using the tables we find that;
Remember the tables give P(X ≤ x) only. Next we shall attempt P(X ≥ 15). Here we shall use P(X ≤ 14) + P(X ≥ 15) = 1. So we shall subtract P(X ≤ 14) from 1 to find P(X ≥ 15) that is;
We use the tables again except this time we find where x=14. We find that;
Note that the tables can also be used to find P(X ≤ x) since
Here is another example. Suppose the random variable X ~ B(25, 0.5). Find;
- P(X ≤ 6)
- P(X = 6)
- P(X > 13)
First we shall start with P(X ≤ 6). Here we use the cumulative binomial tables with n=25, p=0.25 and x=6, we find that;
Next we shall find P(X = 6), we know that;
So we need to find the difference of P(X ≤ 6) and P(X ≤ 5) that is;
Now we shall find P(X > 13). Note the tables give P(X ≤ x) so the alternative is to find P(X ≥ 14) and use the fact that P(X ≥ 14) + P(X > 13) = 1 so we have;
…we can now use the tables to find;
…we find that;
And lastly we need to find P(6 < X ≤ 10). We can think of this expression as P(X = 7, 8, 9, or 10). The interval we’re trying to find is between 6 and 10, so we have;
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.
|…greater than 5…|| X > 5||1 – P(X ≤ 5)|
|…no more than 3…||X ≤ 3P(X ≤ 3)|
|…at least 7…||X ≥ 71 – P(X ≤ 7)|
|…fewer than 10…||X < 10P(X ≤ 9)|
|…at most 8…||X ≤ 8P(X ≤ 8)|
We shall explore some examples below;
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
Let us start by finding the probability that Jane obtains no more than 2 reds.
We first need to define a suitable random variable. This will make it easier to rewrite the question in terms of probabilities and it can help determine the distribution easily, so;
…the distribution is;
… no more than 2 means X ≤ 2, so we get the probability as;
…now we can use the tables to find where n=12, p=0.3 and x=2, we find that we get;
Now let us find the probability of at least 5 reds.
…at least 5 means X ≥ 5, the probability is;