Instructions for using the Binomial Tool
Binomial Probability Distribution: An online, printable lecture.
©Copyright 2000 Tom Malloy
Note: These instructions are abstracted from and can be supplemented by the full web lecture on the Binomial Probability Distribution available through another link on this page.
We will start with a short introduction to the vocabulary and symbols used in the Binomial Tool. These are standard symbols, so you can skip section 1 if you already are familiar with the Binomial Distribution.
1. Introduction to vocabulary and symbols
2. Binomial Tool Instructions
1. Introduction to vocabulary and symbols
Binomial Probability Distribution. When you have a question that asks the probability of a certain number of successes (r) in a certain number (N) of Bernoulli trials then that probability can be calculated from what's called the Binomial Distribution.
Bernoulli Trials
There are processes in nature which are reduced by human operations to two categories and then modeled in terms of probability. This is what we mean by a Bernoulli Trial. A Bernoulli Trial is a process that can have only two outcomes. Common examples of Bernoulli Trials are the flip of a coin (head, tail) and a human child at birth (girl, boy).
Now, as I keep stressing in lectures, most processes are infinite and the two outcomes are a result of our measurement operation. So obviously our child at birth is an infinite process, and the two outcomes exist only because we choose to talk about or do research on gender. If we were doing research on birth weight, the measurement operation would result in a number perhaps in kilograms. There are an infinity of things that might be measured about a human birth and an infinity of things that are not susceptible to measurement. But, let's suppose we are interested in gender. This is fine; but for both human and scientific reasons, it is important to remember the infinity behind the operations.
Success and Failure. Traditionally in the jargon of probability theory one of the two possible outcomes of a Bernoulli Trial is called a success and the other one a failure. There's no value judgment implied by the use of these words in the probability context. Success and failure are being used as conventions. They are merely names which indicate the two outcomes of the Bernoulli Trial. Whatever you decide to call a success and a failure is completely arbitrary.
A head could be called a success. In that case we would say that the probability of a success is .5. Of course that makes a tail a failure. And the probability of a failure would also be .5.
p and q. The probability of a success is typically denoted by a small p. And the probability of a failure is denoted by a small q.
The Binomial Probability Distribution
The Binomial gives you the probability of r successes in N trials.
P(r; p, N). To work with the Binomial we must specify three things: r, p, and N. We must know how many successes we are interested in; we must know the probability of a success: and we must know how many trials we are talking about. The symbols for these three are, of course, r, p, and N. The standard notation for expressing the probability of r successes in N trials with p as probability of a success is P(r: p, N).
For example, suppose we have 8 independent births and we define a success as a girl. Suppose p = .5. We might want to know the probability of 4 girls (successes) in 8 births. This would be expressed as P(4; .5, 8). The probability of 11 girls in 20 births would be P(11; .5, 20).
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Between. Just as with the Normal Distribution, with the Binomial we will distinguish "probabilities between values" from "probabilities outside values." Suppose N = 8, p = .5 and we want to know what the chances are of getting between 2 and 5 girls in 8 births.
The convention is that "between" is inclusive. When I say between 2 and 5 girls I mean 2 or 3 or 4 or 5 girls. Both 2 and 5 are included in the potential number of successes.
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Outside. Conversely, the convention is that the probability of outside 2 and 5 girls excludes 2 and 5. If we have N = 8 births then outside 2 and 5 means 1 or 6 or 7 or 8 births. Both 2 and 5 are excluded.
Between is inclusive. Outside is exclusive.
Binomial Tool Visual Output
The last graphic on this topic shows how the Binomial Distribution looks when drawn by StatCenter's Binomial Tool. We will learn how to use that tool in the next topic.
Notice that the number of successes (r) runs along the horizontal axis. The probability for each number of successes goes up the vertical axis--the higher the black area the higher the probability. And, circled in green, in the top left corner, the standard notation appears. In this case it says P(r; p = 0.5, N = 10).
2. Instructions for Binomial Tool
We will use an example as a basis for step by step instructions. Suppose we flip a fair coin 8 times. Suppose we define a success as a head. Suppose also, the coin is fair, p = .5. The Binomial Distribution allows us to answer questions like what's the probability of 4 heads in 8 flips. Or, what's the probability of between 2 and 5 heads in 8 flips. Or, what's the probability of getting outside 2 and 5 heads in 8 flips. Use StatCenter's Binomial Tool to find these probabilities.
Exactly 4 successes in 8 trials
Define a head as a success, with p = .5. What is the probability of r = 4 successes in 8 trials when p = .5?
Set N. The graphic points where to set the number of trials, N. In this case enter 8, and click the Enter N button.
Set p. The graphic also points out where to set the probability of a success, p. In this case p should already read .5. If not, enter .5 and click the Enter Probability button.
Set Between. Click on the "Between Icon."
Set upper and lower scores. In this case we want to know the probability of exactly r = 4 successes in 8 trials. So we will set both the upper and the lower score to 4. Remember that between is inclusive. So if the both the lower and upper score are set to 4, the probability will include 4 (and only 4). This may seem a bit odd at first, but it works.
Read the Probability. The small white window in the lower right corner should read 0.2734. This is the probability of 4 successes in 8 trials when p = .5. Another way to write this is P(4; .5, 8) = 0.2734.
Black Area. As with the Normal Distribution, with the Binomial the black area represents the relevant probability.
Between 2 and 5 successes in 8 trials
Now let's answer another kind of question. What's the probability of getting between 2 and 5 heads in 8 flips of a fair coin? Here again, you use the binomial tool.
Input information given in the question. Set N = 8, set p = .5, set the lower value of r =2, and set the upper value of r=5. Make sure the Between Icon is clicked.
Then simply read the probability which is .8203. The probability will again be represented by the black area. As usual with both the Normal and Binomial distributions, we're interpreting probability as an area.
Outside 2 and 5 successes in 8 trials
Now let's find the probability of getting outside 2 and 5 heads in 8 tosses of a fair coin. This will be the same of the example we just finished, of course, except that you click on the Outside Icon instead of the Between Icon.
Input information from the question. Click on the Outside Icon. Enter p = .5. Enter N = 8. Enter lower and upper scores of 2 and 5. Remember that "outside" is exclusive and does not include 2 and 5.
Read the probability. In the probability output window in the lower right corner you will find the answer. The probability of getting outside 2 and 5 heads is .1797.
Black Area. Once again the black area represents probability. Click back and forth between the Outside Icon and the Between Icon so you can see the relationship between them both with the black area and the probability. Notice that the probability outside plus the probability between equals 1. That is, .8203+.1797=1.
Play with the Binomial Tool, entering various different parameters to discover what happens. For example, what if N = 100 and p = .5? Or even N = 200 and p = .5. Does the Binomial begin to look like the Normal?