(HP-65) Binomial distribution with large input support

[Dave Britten]

This program calculates the binomial probability mass function, or cumulative binomial probabilities, with support for large (>69) numbers of trials. To do this, it uses Nemes' formula to approximate ln(x!) rather than calculating x! directly and overflowing.

Recommended Key Labels

A: CALC
E: ln(x!)

Usage

There weren't enough steps left for a fancy user interface, so inputs must first be stored directly into R1-R4 before running the program. Store these values:

R1: n - number of trials
R2: p - percentage of success on a single trial
R3: xmin - minimum number of successes
R4: xmax - maximum number of successes

Then press A to calculate the probability mass/cumulative probability. If xmin=xmax, the program calculates the probability mass for that value of x. If xmax>xmin, then the program calculates cumulative probability for the specified range of x values. If xmin>xmax, or xmax>n, you'll get an error.

Note that Nemes' formula is an approximation, which means this program produces approximate results. Accuracy improves with larger inputs, but the worst-case scenario appears to be about 4 digits of accuracy (e.g. n=2, p=0.5, xmin=0, xmax=2 yields about 100.037%). So it's no worse than a slide rule!

Probability mass is calculated in constant time - about 10 seconds - and a cumulative probability with xmax-xmin=10 takes about 30 seconds. Execution time in seconds can be estimated using 10 + 2(xmax - xmin).

Also, you can press E to calculate ln(x!).

Example

If you roll a fair six-sided die 4000 times, what are the odds that you will get any given result between 662 and 672 times? (The mean of this distribution is 666.67.)

4000 STO 1
6 g 1/x STO 2
662 STO 3
672 STO 4
A: 0.1844

There is approximately an 18.44% chance that any specific result will be rolled between 662 and 672 times.

Code:

RCL 1   34 01
E       15
RCL 3   34 03
STO 8   33 08
E       15
-       51
RCL 1   34 01
RCL 3   34 03
-       51
E       15
-       51
RCL 3   34 03
RCL 2   34 02
f       31
LN      07
STO 5   33 05
*       71
+       61
RCL 1   34 01
RCL 3   34 03
-       51
1       01
RCL 2   34 02
-       51
f       31
LN      07
STO 6   33 06
*       71
+       61
f-1     32
LN      07
STO 7   33 07
LSTx    35 00
LBL     23
1       01
RCL 8   34 08
RCL 4   34 04
g x=y   35 23
RCL 7   34 07
RTN     24
g RDown 35 08
RCL 1   34 01
g x><y  35 07
-       51
RCL 8   34 08
1       01
+       61
STO 8   33 08
/       81
f       31
LN      07
+       61
RCL 5   34 05
+       61
RCL 6   34 06
-       51
f-1     32
LN      07
STO     33
+       61
7       07
g LSTx  35 00
GTO     22
1       01
LBL     23
E       15
1       01
+       61
STO 7   33 07
1       01
2       02
*       71
.       83
1       01
RCL 7   34 07
/       81
-       51
g       35
1/x     04
RCL 7   34 07
+       61
f       31
LN      07
1       01
-       51
RCL 7   34 07
*       71
2       02
g       35
PI      02
*       71
RCL 7   34 07
/       81
f       31
LN      07
2       02
/       81
+       61
RTN     24