11/17/2011 AC 23.1309-1E 

Appendix 3 

A3-1 

APPENDIX 3. CALCULATION 

OF THE AVERAGE PROBABILITY PER FLIGHT HOUR 

1. The purpose of this material is to provide guidance for calculating the "average probability 

per flight hour" for a failure condition so that it can be compared with the quantitative 

requirements of § 23.1309. The process of calculating the "average probability per flight hour" 

for a failure condition is a four-step process and based on the assumption that the life of an 

aircraft is a sequence of "average flights." 

Step 1: Determination of the "average flight;" 

Step 2: Calculation of the probability of a failure condition for a certain "average flight;" 

Step 3: Calculation of the "average probability per flight" of a failure condition; and 

Step 4: Calculation of the "average probability per flight hour" of a failure condition. 

 a. Determination of the “average flight:” The "average probability per flight hour" is to be 

based on an "average flight." The applicant should estimate the average flight duration and 

average flight profile for the fleet of aircraft to be certified. The average flight duration should 

be estimated based on the applicant’s expectations and historical experience for similar types. 

The average flight duration should reflect the applicant’s best estimate of the cumulative flight 

hours divided by the cumulative aircraft flights for the service life of the aircraft. The average 

flight profile should be based on the operating weight and performance expectations for the 

average aircraft when flying a flight of average duration in an ICAO standard atmosphere. The 

duration of each flight phase (e.g., takeoff, climb, cruise, descent, approach and landing) in the 

“average flight” should be based on the average flight profile. Average taxi times for departure 

and arrival at an average airport should be considered where appropriate and added to the 

average flight time to obtain "average flight--block time." The average flight duration and 

profile should be used as the basis for determining the "average probability per flight hour" for 

quantitative safety assessment as means of compliance with this AC. 

 b. Calculation of the probability of a failure condition for a certain "average flight:" 

The probability of a failure condition occurring on an "average flight" should be determined by 

structured methods (see ARP 4761 for various methods) and should consider all elements (e.g., 

combinations of failures and events) that contribute to a failure condition. If there is only an 

effect when failures occur in a certain order, the calculation should account for the conditional 

probability that the failures occur in the sequence necessary to produce a failure condition. The 

probabilities of the basic events (component or part level failures) that contribute to the 

probability of a failure condition should consider the following: 

 (1) The individual part, component, and assembly failure rates utilized in calculating the 

"average probability per flight hour" should be estimates of the mature constant failure rates after 

infant mortality and prior to wear-out. Alternatively, a non-constant failure rate can be used (i.e. 

Weibull or other accepted means). Inspection intervals or component life limits employed to 

protect against wear out are to be placed in chapter 4 or 5 of the maintenance manual. In either 

case, the failure rate should be based on all causes of failure (operational, environmental, etc.). 

Where available, service history of same or similar components in the same or similar 

environment should be used. 
11/17/2011 APPENDIX 3 (CONTINUED) AC 23.1309-1E 

 Appendix 3 

A3-2 

 (2) If the failure is only relevant during certain flight phases, the calculation should be 

based on the probability of failure during the relevant "at risk" time for the "average flight." 

 (3) If one or more failed elements in the system can persist for multiple flights (latent, 

dormant, or hidden failures), the calculation has to consider the relevant exposure times (e.g., 

time intervals between maintenance checks/ inspections). In such cases, the probability of the 

failure condition increases with the number of flights during the latency period. 

 (4) If the failure rate of one element varies during different flight phases, the calculation 

should consider the failure rate and related time increments in such a manner as to establish the 

probability of the failure condition occurring on an "average flight." It is assumed that the 

"average flight" can be divided into n phases (e.g., phase 1, ... , phase n). Let TF the "average 

flight" duration, Tj the duration of phase j and tj the transition point between Tj and Tj+1, j = 

1, ... , n . I.e. 

Let j(t) the failure rate function during phase j, i.e. for t  [tj-1,tj]. 

Remark: j(t) may be equal 0 for all t  [tj-1,tj] for a specific phase j. 

Let PFlight(failure) the probability that the element fails during one certain flight 

(including non-flying time) and PPhase j(failure) the probability that the element fails 

in phase j. 

Two cases are possible: 

 

(i) The element is checked operative at the beginning of a certain flight. Then 

F

j=1

n

T T = =  j j j-1 j and t - t T ; j = 1, ... , n

   

   

 















   

   



n

i=1

t

t

i

n

j 1

j 1 j

n

j=1

Flight Phase j

i

i-1

1 exp d

P (failure) P (failure) P failure t t ,t

 x x
11/17/2011 APPENDIX 3 (CONTINUED) AC 23.1309-1E 

 Appendix 3 

A3-3 

(ii) The state of the item is unknown at the beginning of a certain flight. Then 

where Pprior (failure) is the probability that the failure of the element has occurred 

prior to a certain flight. 

Note: For the two mathematical operators, ∏ is a product sign and ∈ is element of. 

 (5) If there is only an effect when failures occur in a certain order, the calculation should 

account for the conditional probability that the failures occur in the sequence necessary to 

produce a failure condition. 

c. Calculation of the “average probability per flight” of a failure condition: The next 

step is to calculate the "average probability per flight" for a failure condition, that is, the 

probability of a failure condition for each flight (which might be different, although all flights 

are "average flights") during the relevant time (for example, the least common multiple of the 

exposure times or the aircraft life) have to be calculated, summed up, and divided by the number 

of flights during that period. The principles of calculating are described below and are in more 

detail in ARP 4761. 

 

Note: N is the number of all flights during the relevant time, and PFlight k is the probability that a 

failure condition occurs in flight k. In the special case of a duplex system (i.e., one component 

failure latent, the other detected), this method results in an "average probability per flight," 

which equals the product of both failure rates multiplied by the "average flight" duration TF

multiplied by one-half (50 percent) of the relevant exposure time.

 d. Calculation of the “average probability per flight hour” of a failure condition: Once 

the "average probability per flight" is calculated, it should be normalized by dividing it by the 

"average flight" duration TF in “flight hours” to obtain the "average probability per flight hour." 

This quantitative value should be used in conjunction with the hazard category/effect established 

by the FHA to determine if it is compliant for the failure condition being analyzed. 

    































      n

i=1

t

t

prior i

Flight prior

i

i-1

1 P (failure) 1 exp d

P (failure) = P (failure)

 x x

 

 

N

k=1

P failure condition

P failure condition

N

Flight k

Average per Flight





   

F

Average per Flight

Average per FH T

P failure condition

P failure condition 