Mathematics · Probability
At Least One Event from Failure Probabilities failure probability for second independent event Solver
Rearrange the at least one event from failure probabilities relationship and solve for failure probability for second independent event.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Use b=(1−c)/a with probability of at least one success=0.44000000000000006 and failure probability for first independent event=0.7.
- failure probability for second independent event=0.7999999999999999.
- Substitution into c=1−ab reconstructs 0.44000000000000006.
Understand At Least One Event from Failure Probabilities: solve failure probability for second independent event
One idea, three depths
Choose how deeply to explain At Least One Event from Failure Probabilities: solve failure probability for second independent event
At Least One Event from Failure Probabilities: solve failure probability for second independent event: Rearrange the at least one event from failure probabilities relationship and solve for failure probability for second independent event.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using At Least One Event from Failure Probabilities: solve failure probability for second independent event to answer this question: rearrange the at least one event from failure probabilities relationship and solve for failure probability for second independent event? Enter probability of at least one success and failure probability for first independent event; the calculator shows failure probability for second independent event. For example: failure probability for first independent event=0.7 and failure probability for second independent event=0.8 produce probability of at least one success=0.44000000000000006. The answer tells you failure probability for second independent event.
Age 15Explain it to a 15-year-oldConnect it to the formula
For two independent trials, one minus the probability that both fail gives the probability of at least one success. This page isolates failure probability for second independent event and verifies it in the original relationship. The rule is b=(1−c)/a. Its input values are probability of at least one success, failure probability for first independent event, and the main result is failure probability for second independent event. For example: failure probability for first independent event=0.7 and failure probability for second independent event=0.8 produce probability of at least one success=0.44000000000000006.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated at least one event from failure probabilities: solve failure probability for second independent event relation over the valid real-number domain stated below. The implemented relation is b=(1−c)/a, evaluated from probability of at least one success, failure probability for first independent event to produce failure probability for second independent event. For two independent trials, one minus the probability that both fail gives the probability of at least one success. This page isolates failure probability for second independent event and verifies it in the original relationship. The inputs are failure probabilities and independence must be justified.
Inputs and valid domain
- probability of at least one success must be a finite real number.
- failure probability for first independent event must be a finite real number.
Important boundary: The inputs are failure probabilities and independence must be justified.
The formula
b=(1−c)/a
How the calculator works through it
It substitutes probability of at least one success, failure probability for first independent event into the formula and exposes every numerical step above. The main output is failure probability for second independent event, accompanied by Reconstructed probability of at least one success.
Read the result correctly
The failure probability for second independent event is the direct answer to “rearrange the at least one event from failure probabilities relationship and solve for failure probability for second independent event.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
failure probability for first independent event=0.7 and failure probability for second independent event=0.8 produce probability of at least one success=0.44000000000000006.
Where this model stops being reliable
The inputs are failure probabilities and independence must be justified.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why At Least One Event from Failure Probabilities: solve failure probability for second independent event works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
At Least One Event from Failure Probabilities: solve failure probability for second independent event uses b=(1−c)/a. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Probability as a modelled proportion
Probability rules are needed to interpret what the At Least One Event from Failure Probabilities: solve failure probability for second independent event result says about possible outcomes.
Review this foundation about 5 min
Optional enrichment
- Ordered arrangements
Counting ordered arrangements can extend At Least One Event from Failure Probabilities: solve failure probability for second independent event to more detailed sample spaces and event models.
Review this foundation about 5 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read probability of at least one success, failure probability for first independent event.
- Evaluate the principal relationship: b=(1−c)/a.
- Return failure probability for second independent event and check the domain conditions described above.
Python
from math import *
def at_least_one_from_failure_probabilities_solve_b(c, a) -> float:
return ((1.0 - c) / a)
assert abs(at_least_one_from_failure_probabilities_solve_b(0.44000000000000006, 0.7) - 0.7999999999999999) < 1e-6 * max(1.0, abs(0.7999999999999999))
C
#include <assert.h>
#include <math.h>
double at_least_one_from_failure_probabilities_solve_b(double c, double a) {
return ((1.0 - c) / a);
}
int main(void) {
const double expected = 0.7999999999999999;
const double actual = at_least_one_from_failure_probabilities_solve_b(0.44000000000000006, 0.7);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double at_least_one_from_failure_probabilities_solve_b(double c, double a) {
return ((1.0 - c) / a);
}
int main() {
constexpr double expected = 0.7999999999999999;
const double actual = at_least_one_from_failure_probabilities_solve_b(0.44000000000000006, 0.7);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double at_least_one_from_failure_probabilities_solve_b(double c, double a)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global at_least_one_from_failure_probabilities_solve_b
section .text
at_least_one_from_failure_probabilities_solve_b:
push rbp
mov rbp, rsp
sub rsp, 48
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
mov rax, 0x3ff0000000000000
movq xmm0, rax
movsd [rbp-40], xmm0
movsd xmm0, [rbp-40]
subsd xmm0, [rbp-8]
movsd [rbp-32], xmm0
movsd xmm0, [rbp-32]
divsd xmm0, [rbp-16]
movsd [rbp-24], xmm0
movsd xmm0, [rbp-24]
leave
ret
MATLAB
function result = at_least_one_from_failure_probabilities_solve_b(c, a)
result = ((1.0 - c) / a);
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[c_, a_] := ((1.0 - c) / a);
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Introductory Statistics 2e
Read the free OpenStax statistics textbookCite this book
- APA 7
- Illowsky, B., & Dean, S. (2023). Introductory statistics 2e. OpenStax. https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
- MLA 9
- Illowsky, Barbara, and Susan Dean. Introductory Statistics 2e. OpenStax, 2023, https://openstax.org/books/introductory-statistics-2e/pages/1-introduction.
- Chicago author-date
- Illowsky, Barbara, and Susan Dean. 2023. Introductory Statistics 2e. Houston, TX: OpenStax. https://openstax.org/books/introductory-statistics-2e/pages/1-introduction.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). At Least One Event from Failure Probabilities failure probability for second independent event Solver. MW SysArc Tools. https://math.mwsysarc.com/probability/at-least-one-from-failure-probabilities-failure-probability-for-second-independent-event-solver
MLA 9
MW SysArc. “At Least One Event from Failure Probabilities failure probability for second independent event Solver.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/probability/at-least-one-from-failure-probabilities-failure-probability-for-second-independent-event-solver. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “At Least One Event from Failure Probabilities failure probability for second independent event Solver.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/probability/at-least-one-from-failure-probabilities-failure-probability-for-second-independent-event-solver.
Harvard
MW SysArc (2026) ‘At Least One Event from Failure Probabilities failure probability for second independent event Solver’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/probability/at-least-one-from-failure-probabilities-failure-probability-for-second-independent-event-solver (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_at_least_one_from_failure_probabilities_solve_b_2026,
author = {{MW SysArc}},
title = {At Least One Event from Failure Probabilities failure probability for second independent event Solver},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/probability/at-least-one-from-failure-probabilities-failure-probability-for-second-independent-event-solver},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - At Least One Event from Failure Probabilities failure probability for second independent event Solver
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/probability/at-least-one-from-failure-probabilities-failure-probability-for-second-independent-event-solver
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the At Least One Event from Failure Probabilities: solve failure probability for second independent event do?
Rearrange the at least one event from failure probabilities relationship and solve for failure probability for second independent event.
How does the At Least One Event from Failure Probabilities: solve failure probability for second independent event work?
The calculator applies b=(1−c)/a. For two independent trials, one minus the probability that both fail gives the probability of at least one success. This page isolates failure probability for second independent event and verifies it in the original relationship.
What can I learn from the At Least One Event from Failure Probabilities: solve failure probability for second independent event?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .