Binomial Probability Calculations on the TI-85
As far as I can tell, the TI-85 lacks the probability-density and cumulative-distribution
function commands related to binomial probabilities. You can calculate
ranges of binomial probabilities, as in r = 4 to 9 or r < 6, using the
sequence and summation commands however. Furthermore, to limit the amount
of typing one needs to perform, you can store a template command to an
equation variable and simply execute that single command (once appropriate
values have been stored in the template variable parameters).
A Sample Calculation: Suppose we have a binomial distribution
with n = 25 trials and a probability of success given by p = 0.3. We want
to calculate the probability that we witness from 4 to 8 successes (inclusive).
That corresponds to adding up the numbers "n choose r" times p^r times
(1 - p)^(n - r) from r = 4 to 8. The sequence command, seq, accessed
by pressing [2nd] [LIST] [F5: OPS] [MORE] [F3: seq], can be used
to quickly calculate the values to be added. The summation command, sum,
accessed by pressing [2nd] [LIST] [F5: OPS] [MORE] [F1: sum], will
then add them up. The command
seq(f(x),x,a,b,dx)
will evaluate the formula f(x) by substituting all values of x from a to
b with steps of size dx. These values are given as a list or sequence.
The sum command is given a list of numbers and it simply adds them,
thus sum({1,1,2}) returns the value 1 + 1 + 2 = 4. Our sample
calculation can then be carried out with the command
sum(seq( (25 nCr x)*.3^x*.7^(25-x), x, 4, 8, 1) )
If you do this (recall that nCr is located under [MATH] [F2:Prob]
[F3: nCr] ), you should get 0.643687 or so.
A Slightly Better Method: Much of the command needed to calculate
sums of binomial probabilities can be stored in an equation variable. One
then sets values of the parameters (n, p, where to start, where to stop)
and executes a single command to calculate these probabilities slightly
quicker. Unfortunately, there are four parameters which need setting, so
this amounts to a fair bit of typing as well. You can enter the following
command (exactly, paying attention to lowercase-uppercase letters) at the
home screen, then store values in the variables N (number of trials), P
(probability of a success), A (the left endpoint of the interval) and B
(the right endpoint of the interval) and finally evaluate the equation
variable BP to get the answer. Here is the command to save the equation
(BP for binomial probability). Type:
BP=sum(seq( (N nCr x)*P^x*(1 - P)^(N - x),x,A,B,1) )
and then press [ENTER]. This equation is now stored in your calculator's
memory (forever, or at least until you delete it). To re-calculate the
probability in the last example, issue the commands below (which store
the appropriate values in the parameters).
25 [STO>] N
0.3 [STO>] P
4 [STO>] A
8 [STO>] B
You can now calculate the probability that a binomial random variable with
p = 0.3 has 4 to 8 successes in 25 trials by simply pressing BP [ENTER].
Note, you can let A = B to calculate a single binomial probability.
For example, if you roll 30 dice and want to know the likelihood that you
obtain exactly seven 4's, issue the commands:
30 [STO>] N
1 / 6 [STO>] P
7 [STO>] A
7 [STO>] B
PB
You should get the answer 0.1098 (or so).