Stella Academy

mathematics Module

Interactive Simulation Screen

SYS_OK

Lesson Directive // Mission ProbabilityREF_CORE

The Risk of Failure

P_{sys}

=

1

-

(1

-

p

)^n

Hover over a variable in the formula above, or see glossary below:

P_{sys}

System Reliability

p

Component Reliability

n

Redundant Components

Count

In spaceflight, components fail. If a critical system relies on a single component with a 90% success rate, there is a 10% chance of complete mission failure. We express these chances as Fractions or decimals (e.g. 0.10).

INSIGHT: For single-point failures, the system reliability equals the component reliability.

Adding Redundancy

To increase reliability, engineers add backup components in parallel. The system only fails if ALL of the redundant components fail at the exact same time.

INSIGHT: The probability of all components failing is the failure rate of one component multiplied by itself n times.

Calculating System Success

First, calculate the probability of a single failure (1 - p). Then, calculate the probability that all n components fail: (1 - p)^n. Finally, subtract that from 1 to find the probability of at least one success.

INSIGHT: Even with low-reliability parts, high redundancy can create an ultra-reliable system.

●Telemetry Control Console

CAL_SYS

Adjust Simulator Parameters

Syllabus Worked Example

Q: A spacecraft's primary flight computer has a reliability of 0.90 (90%). If the spacecraft has n = 3 identical, independent computers, what is the overall system reliability (P_{sys})?

1.Find the failure rate of one computer: 1 - 0.90 = 0.10.

2.Calculate the probability that ALL three fail: 0.10^3 = 0.10 \times 0.10 \times 0.10 = 0.001.

3.Subtract the total failure probability from 1 to find the success probability: 1 - 0.001.

Result:P_{sys} = 0.999 \text{ (99.9\%)}