NDRC Small Target Bombing Probabilities Calculator

(Created June 2024)

During WWII, the Applied Mathematics Panel of the U.S. National Defense Research Committee (NDRC) produced a small number of specialized slide rules for use by operations analysts. One of these slide rules was the Small Target Bombing Probabilities computer.

A copy of the STBP computer was obtained by John Ptak of the JF Ptak Science Books website (LINK) and semi-reproduced online.

You can view the scans that Mr. Ptak made here (FRONT and BACK) along with a different set of scans (FRONT and BACK).

A small write up of the computer was contained within the Summary Technical Report of the Applied Mathematics Panel, NDRC - Volume 3: Probability and Statistical Studies in Warfare Analysis (8.9 MB PDF / Library of Congress Online Version), and the writeup was:

A slide rule, titled Small-Target Bombing Probabilities, is available for calculation of the expected number E of hits and the probabilities Pk of at least k hits (k = 1, 2, 3, 4) for attacks on small targets.

The important assumption made in designing the slide rule is:

That in a region around the origin, of radius about half that of the circular probable error CEP circle, the distribution is statistically uniform, the density being equal to the central density of a circular-Gaussian distribution. The answers of the slide rule are correct for a small centrally located target when the aiming-error statistic, CEP, is estimated on the assumption that the distribution is strictly Gaussian.

The slide rule, in effect, computes the probability that (1) a bomb will fall in this central region, and (2) it will hit a target in that region. The rule is shown in Figure 6.


Original Image
2x Upscaled Image

The slide rule for small-target bombing probabilities has been manufactured in small quantities and distributed to some operations analysts and other personnel in the Services.

The instructions on the back of the computer were:

Small Target Bombing Probabilities

Purpose

This calculator computes the expected number of hits, and the probability of at least a certain number of hits, when a small target is attacked by a number of bombers. The results obtained will be correct only when the manner of bombing is such that the impacts from the various trains of bombs intermingle to form one general pattern, and when the circular probable error (CEP) of this pattern is at least as large as the greatest dimension of the target.

Instructions

1. Turn the larger disc until the arrow on this disc indicates the NUMBER OF BOMBS to be released.

2. Turn the smaller disc until its arrow indicates the AIMING ERROR IN FEET on the larger disc.

3. Turn the transparent radial index into alignment with the AREA OF TARGET IN THOUSANDS OF SQUARE FEET (on the smaller disc).

4. The radial index now indicates, on the larger disc, the EXPECTED NUMBER OF HITS PER THOUSAND BOMBS.

5. The radial index also indicates, on the outer scales, (a) the EXPECTED NUMBER OF HITS for the indicated number of bombs, (b) the PROBABILITY OF AT LEAST ONE HIT, (c) the PROBABILITY OF AT LEAST TWO HITS, (d) the PROBABILITY OF AT LEAST THREE HITS, (e) the PROBABILITY OF AT LEAST FOUR HITS.

Illustration

What is the probability of hitting a rectangular target 70 feet wide by 1000 feet long, if 200 bombs are released and 50% of them can be expected to hit within 1200 feet of the center of the target? How many bombs should be released if the probability must be 0.80?

The answer to the first question is obtained as follows: (1) set the arrow on the larger disc to 200 bombs; (2) set the arrow on the smaller disc to an aiming error radial index to 70,000 square feet (= 70 x 1000 = the area of the target); (4) then, from the various scales, it is found that: (a) 10 hits are expected per thousand bombs, (b) 2 hits are expected from the 200 bombs, (c) the probability of at least one hit is 0.87, and (d) the probability of at least two hits is considerably less than 0.75.

To answer the second question, turn the two discs and the radial index as a unit until the radial index indicates a probability of .80 of at least two hits; if the other settings have not been disturbed, it will be found that nearly 300 bombs are required.

Using Chat GPT 4σ, I was able to recreate to a fair level of accuracy the original equations used for this computer and implement them on this webpage.

Small Target Bombing Probability Calculator

NOTE: You can enter larger numbers with commas, e.g. 5,000,000 - the program will automatically strip the commas. Likewise, the program will automatically recognize the following SI prefixes:

  • K - Kilo (103)
  • M - Mega (106)

Hence, a 1 million ft2 target would be power level would be "1M", while a 65,000 ft2 target would be "65K".

Number of Bombs Dropped
  • USN VB (SBD) Squadron: 12 a/c, 1 x 1000 lb per a/c, 12 per squadron
  • USN VTB (TBF) Squadron: 18 a/c, 4 x 500 lb per a/c, 72 per squadron
  • AAF B-17 Bomb Group: 36 a/c; 12 x 500 lb per a/c, 432 per group
Area of Target (ft2)
  • Gato Class SS: 311 x 27 ft = 6,247 ft2
  • Fletcher Class DD: 376 x 39 ft = 12,470 ft2
  • Liberty (EC2) Ship: 441.5 x 56.89 ft = 23,056 ft2
  • Cleveland Class CL: 610 x 66 ft = 28,868 ft2
  • T2-SE-A1 Tanker: 523 x 68 ft = 31,636 ft2
  • Baltimore Class CA: 670 x 70 ft = 36,142 ft2
  • Iowa Class BB: 887 x 108 ft = 75,205 ft2
  • Essex Class CV: 870 x 89 ft (flight deck) = 77,430 ft2
  • Detroit Tank Arsenal: 1800 x 595 ft = 1,071,000 ft2
  • Pentagon: Five Sided Pentagon, 921 ft each = 1,304,228 ft2
  • WW2 AAF 1,000 ft Radius Circle: 3,140,000 ft2
Aiming Error (ft)
  • SBD Dauntless: 175-200 ft CEP
  • B-17 1943: 1200 ft CEP

Results

The key equation(s) used by the calculator are:

Expected Hits per 1000 Bombs

Where:

H1000: Expected Hits per Thousand Bombs.
TArea: Target Area (square feet)
CEP: Circular Error Probable (CEP) (feet)

Expected Hits from B Bombs

Where:

HB: Expected Hits from B number of bombs.
H1000: Expected Hits per Thousand Bombs.
B: Number of bombs dropped.

The chances of n amount of bomb hits occuring are calculated via the:

Poisson Cumulative Probability Formula

Where:

λ (lambda): is set to HB -- Expected Hits from B number of bombs.
e: Natural Log = 2.71828.
i: Index of cumulative event you wish to calculate. If you want to find out the probability of at least one bomb hitting the target area, you set this to 0. If you want to find the probability of a second bomb, you set the index to 1, and so on.