## Binomial Probability Distribution

The Binomial Probability Function can be used to calculate the probability of "x" successes from "n" trials with the probability of success on each trial "p". We say "x", the random variable, is distributed Binomial(n,p,x).

Sample Problem
Say we want to calculate the probability of 3 successes from a total of 10 independent trials where the probability of success on each trial is 0.35. Here : "n" = 10 ,"x" = 3 , and "p"= 0.35 .

On the calculator start from a clear screen. Press (you will see this screen)

Arrow down to choice "0: binompdf ( " Press .
(you will see this screen.)

You must enter n , p , x in that order inside the parenthesis. So for our problem we will enter 10 .35 . Don't forget the commas or the closed parenthesis!
Now press .252219625 will be displayed. This is the probability of 3 successes from 10 trials where the probability of success on each trial is 0.35 .

What if you wanted to know the probability of 5 successes from this same distribution?
You do not have to start over from the beginning.

To calculate the probability of 5 successes from the same problem you do not have to start over. Simply press
(this screen will reappear.)

Now you can Arrow left to highlight the 3 . Next press . This will replace the 3 with a 5. Press .1535704107 will be displayed. This is the probability of 5 successes from 10 trials where the probability of success on each trial is 0.35.

"Binomcdf(" will calculate the cumulative probability from 0 through X. [e.g. Binomcdf (10 , .20 , 4) will calculate the probability of x = 0+1+2+3+4 success, where n=10, p=0.20, which = .9672065025 ]

Plot a particular Binomial Distribution.

Let's use this last problem's data where "n"= 10, "p"= 0.35. First press
"EDIT" should be highlighted. Press
Make sure all list are clear. [ To clear L1 you Arrow up to and highlight L1, press . Follow the same procedure to clear any list.]

Enter the numbers {1,2,3,4,5,6,7,8,9,10} into List 1.
These are the "x" values. Next we want to enter a formula into List 2.

First highlight L2. You should see this screen.

Now press 10 MATH
Next Arrow left to "PRB"
highlight "3: nCr"
Press . (this screen will appear)

Next press .35 .
You will see this text scroll across the bottom of the screen:

Now press .65 10
You will be able to see all the text by Arrow left or right .
Now press These probabilities will appear in List2. (See below)

[The numbers in List2 are the probabilities associated with the "x" values from List1.] (i.e. The probability that x = 5 is 0.15357, etc. )

Now to see a plot we must go to the STAT PLOT so press Y =
(You will see this screen.)

"1:Plot1..." should be highlighted. Press Make sure On is highlighted and press . (you should be looking at this screen.)

Arrow down and highlight the first option right of "Type" (Scatter Plot)
And press Make sure Xlist: L1 , Ylist: L2

Now, as always, we need to adjust the window to fit our data or you may press to have the calculator automatically adjust the window.

Press Xmin: 0 , Xmax: 11 , Xscl 1 , Ymin: 0 , Ymax: 0.5 , Yscl 0.1 .

Press . (this screen will appear)

Notice that even with only 10 data points this graph resembles a "bell curve."

You can read the probabilities directly off this screen by pressing and Arrow right or Left. (see screen)

Here "x" is the number of successes and "y" is the corresponding probability. So from this screen we see that there is a .2377 chance of 4 successes from 10 trials where the probability of success on each trial is 0.35.

[You can Arrow left or right to read each desired probability.] Notice also that in the upper left screen P1: L1,L2 is displayed. This tells us that we are using PLOT1 and the data from LIST1, and LIST2.