Probability

This program computes a Poisson probability distribution for a user-defined number of occurrence categories (3 to 12) per unit of time. The user supplies the maximum occurrences per time unit, the total number of occurrences, and the total time; from these the mean rate D = B/C is derived. The program approximates e using the classic limit (1 + 1/B)^B rather than a built-in constant, then computes each Poisson term P(k) = e^(−D) · D^k / k! up to the chosen maximum. A correction factor N is formed by summing all computed terms so that the displayed percentages are normalised to the truncated distribution, and each percentage is rounded to two decimal places using the INT(x + 0.5) idiom. A running total is printed at the end of line 570 as a checksum.

Program Structure

The program is divided into four broad phases:

1. Initialisation (lines 9–22): CLEAR and zeroing of all accumulator variables (S through BB).
2. Input (lines 25–95): Three prompts collect the maximum occurrence count A (range 3–12), total occurrences B, and total time C.
3. Computation (lines 100–366): Mean rate, Poisson terms, normalisation sum, and rounded percentage values are calculated.
4. Output (lines 370–570): String labels are assigned and all results printed in a single long PRINT statement.

Mathematical Approach

The Poisson probability mass function P(k) = e^−λ · λ^k / k! is computed term by term. The mean rate λ is D = B/C. Rather than using a stored constant for e, line 110 uses:

LET E=(1+(1/B))**B

This is the well-known limit definition of e, evaluated with B (total occurrences) as the large integer — a reasonable approximation when B is large, but it introduces compounding error for small B. The base probability F = 1/E^D (line 120) is e^−D, and each subsequent term multiplies by powers of D divided by the factorial of k, hard-coded through 12! = 479001600 (line 237).

Normalisation and Rounding

Because the distribution is truncated at the user-chosen maximum, the sum of all computed raw probabilities N (line 255) will be less than 1.0 unless the tail beyond A is negligible. Each percentage is corrected by dividing by N before rounding:

LET PP=(INT (P*10000/N+0.5))/100

This is the standard Sinclair BASIC round-to-two-decimal-places idiom: multiply by 100× the desired precision, add 0.5, truncate with INT, then divide back. The final item in the PRINT at line 570 sums all percentage values as a checksum; due to rounding it may not equal exactly 100.

Conditional Term Computation

Lines 160–237 use a staircase of IF A>=n THEN LET … guards so that only terms up to the user-requested maximum are computed. Variables for unused terms remain at zero (set at lines 11–22), which is essential because they all enter the normalisation sum N at line 255 regardless.

Variable Naming

The program exhausts all single-letter numeric variables and must fall back to two-letter names (AA, BB) for k = 11 and k = 12, and three-letter names (AAA, BBB) for the corresponding percentages.

Notable Techniques and Anomalies

• The e approximation at line 110 reuses the input value B as the iteration count. This is unconventional and ties numerical accuracy to the data rather than using a fixed large integer or a stored constant.
• Line 32 warns “RANGE ALLOWED 3 TO 12” but there is no validation; entering A < 3 merely computes a reduced distribution (only k = 0, 1, 2 are always computed unconditionally via P, Q, R). Entering A > 12 is silently ignored, capping results at 12.
• All 13 labels and values are printed in one very long PRINT statement at line 570, which is syntactically valid but unusual and makes the line difficult to edit in the BASIC editor.
• The REM at line 252 is the only comment in the program, labelling the normalisation step as a “CORRECTION FACTOR”.
• Factorial denominators are entered as literal integers (e.g., 39916800 for 11!, 479001600 for 12!), which is correct but would overflow a 32-bit integer — here they are used as floating-point literals, so precision is adequate.

   9 CLEAR 
  10 PRINT "  LONG TERM PROBABILITY STUDY"
  11 LET S=0
  12 LET T=0
  13 LET U=0
  14 LET V=0
  15 LET W=0
  16 LET X=0
  17 LET Y=0
  18 LET Z=0
  20 LET AA=0
  22 LET BB=0
  25 PRINT 
  30 PRINT "INPUT THE MAXIMUM OCCURRENCES IN ANY ONE UNIT OF TIME "
  31 PRINT 
  32 PRINT "      RANGE ALLOWED 3 TO 12"
  40 INPUT A
  45 CLS 
  50 PRINT "INPUT TOTAL OCCURENCES"
  60 INPUT B
  70 PRINT 
  80 PRINT "INPUT TOTAL TIME"
  90 INPUT C
  95 CLS 
 100 LET D=B/C
 110 LET E=(1+(1/B))**B
 120 LET F=1/(E**D)
 130 LET P=F
 140 LET Q=D*F
 150 LET R=D**2/2*F
 160 IF A>=3 THEN LET S=D**3/6*F
 170 IF A>=4 THEN LET T=D**4/24*F
 180 IF A>=5 THEN LET U=D**5/120*F
 190 IF A>=6 THEN LET V=D**6/720*F
 200 IF A>=7 THEN LET W=D**7/5040*F
 210 IF A>=8 THEN LET X=D**8/40320*F
 220 IF A>=9 THEN LET Y=D**9/362880*F
 230 IF A>=10 THEN LET Z=D**10/3628800*F
 233 IF A>=11 THEN LET AA=D**11/39916800*F
 237 IF A=12 THEN LET BB=D**12/479001600*F
 240 PRINT "PROBABILITY OF","PERCENT"
 250 PRINT 
 252 REM CORRECTION FACTOR
 255 LET N=P+Q+R+S+T+U+V+W+X+Y+Z+AA+BB
 260 LET PP=(INT (P*10000/N+0.5))/100
 270 LET QQ=(INT (Q*10000/N+0.5))/100
 280 LET RR=(INT (R*10000/N+0.5))/100
 290 LET SS=(INT (S*10000/N+0.5))/100
 300 LET TT=(INT (T*10000/N+0.5))/100
 310 LET UU=(INT (U*10000/N+0.5))/100
 320 LET VV=(INT (V*10000/N+0.5))/100
 330 LET WW=(INT (W*10000/N+0.5))/100
 340 LET XX=(INT (X*10000/N+0.5))/100
 350 LET YY=(INT (Y*10000/N+0.5))/100
 360 LET ZZ=(INT (Z*10000/N+0.5))/100
 363 LET AAA=(INT (AA*10000/N+0.5))/100
 366 LET BBB=(INT (BB*10000/N+0.5))/100
 370 LET A$="0 OCCURRENCES"
 380 LET B$="1      """
 390 LET C$="2      """
 400 LET D$="3      """
 410 LET E$="4      """
 420 LET F$="5      """
 430 LET G$="6      """
 440 LET H$="7      """
 450 LET I$="8      """
 451 LET J$="9      """
 452 LET K$="10     """
 455 LET L$="11     """
 457 LET M$="12     """
 570 PRINT A$,PP,B$,QQ,C$,RR,D$,SS,E$,TT,F$,UU,G$,VV,H$,WW,I$,XX,J$,YY,K$,ZZ,L$,AAA,M$,BBB,,,,PP+QQ+RR+SS+TT+UU+VV+WW+XX+YY+ZZ+AAA+BBB