TI-83 Users Guide

Binomial Probability Distributions

__ BINOMIAL EVENTS:__ Binomial events are harder to define than to explain, so here are some examples:

1. A basketball player who is a 60% free throw shooter attempts ten free throws in a game.

2. A family has five children with the probability of 1/2 that any child is a boy.

3. John randomly guesses at the last 6 multiple choice questions on the test knowing the probability of getting a correct answer is 1/4.

Each of these situations has something (assumed) to be in common. Each involves an event in which the outcome is assumed to be random [shooting the ball, having a child, guessing an answer]. The event is repeated [ten shots, five children, six questions]. The number of times the event is repeated is usually symbolized by the letter n, as in n=10. The probability of "success" on any try is the same [60%, 1/2, 1/4] and not affected by the previous tries (independent). The probability is usually symbolized by the letter p, as in p=1/4.

__ EVALUATING BINOMIAL PROBABILITIES: __ The TI-83 allows you to compute the individual probability of a binomial outcome, or the complete distribution. The primary keys for working with binomial events are the "

Here are several ** examples** and the display to compute them:

A) Find the probability that a 60% free throw shooter will make EXACTLY six of eight free throws. In this situation n=8, p=.6, and we want to find the probability of an exact number of successes, k=6. The entry, and the solution, is shown in figure 1. Notice that the probability of approximately 21% is the probability of this exact event. If we wanted the probability that they would score *six or less, *then we would use the cumulative frequency distribution as shown in the second figure. This is the probability that he will hit zero or one or two or … etc up to, and including, six shots. The probability that he will hit MORE than six, is the complement of this outcome. 1- .89362432

If we are interested in the probability of two or more different outcomes, we can put them in a list after the entry of p. To compute the probability that the shooter in the previous situation makes five, six or seven free throws we add the list {5,6,7} to the end. The calculator returns a list with the three individual probabilities. [* Note that we could not see the whole list because of the long decimals, so we set the decimal length under the MODE key*

__ A BINOMIAL DISTRIBUTION:__ Sometimes we want to look at the probabilities of ALL the possible outcomes, and even setting the decimal length will not allow us to see all of them easily. It is at such a time that we utilize the list capabilities of the TI-83. Let's explore the possibilities of John's guesses on the multiple choice test. To expand the information, lets use List1 to store the list of possible outcomes. For six questions he could get any number from zero to six correct, so we want to put these integers in the list. The screen display below left shows one way to do this. Now we use a simple command to display the probability of each outcome in List2. This command is shown in the second screen below. Note that NO k value was used, and so the entire possible list of probabilities was returned.

The results displayed in the STAT/EDIT screen are shown at right above. This list just fits, but if it were longer we could scroll through it to check possible outcomes. A table of outcomes and related probabilities such as the one we have created above is called a __Probability Density Table.__

__ GRAPHICS AND STATISTICS:__ Having created a Probability Density Table, it seems only natural to want to Display a Graph of the results. To do so is not too difficult after we have come so far. Press

Finally, we may ask ourselves about the statistics involved in John's efforts to achieve academic success through randomized guesses. From the home screen, press the [**STAT]** key and select the **calc** menu. Then press enter for 1-Var Stats. The home screen will display the image at left below. Add the instructions L1,L2 to indicate that the events are in L1 and the probabilities are in L2. The completed command is in the middle screen below.

When you press ** **the calculator will display the standard statistics associated with this event, as shown at the right above. Notice that the average number of questions correct by this process is 1.5, which is what we should expect out of 6 questions with a probability of 1/4.

__ TRY THESE:__ Now it is your turn. We have avoided the question of the number of male children born in families of five so that you would have a practice problem. Try to answer each of these questions about the situation.

1) What is the probability that exactly three of the children are boys?

2) What is the probability that less than three were boys?

3) What is the probability that there were three or four boys.

4) Construct a Frequency Distribution Table for the situation.

5) Draw the Frequency Distribution Polygon for the situation.

6) Find the mean and standard deviation of the number of boys born in families of five.