---
title: "Return period"
chunk: 2/2
source: "https://en.wikipedia.org/wiki/Return_period"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T03:51:25.813539+00:00"
instance: "kb-cron"
---

== Binomial ==
In a given period of 
  
    
      
        n
        ×
        τ
      
    
    {\displaystyle n\times \tau }
  
 for a unit time 
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
 (e.g. 
  
    
      
        τ
        =
        1
        
          year
        
      
    
    {\displaystyle \tau =1{\text{year}}}
  
), the probability of a given number r of events of a return period 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 is given by the binomial distribution as follows.

  
    
      
        P
        (
        X
        =
        r
        )
        =
        
          
            
              (
            
            
              n
              r
            
            
              )
            
          
        
        
          μ
          
            r
          
        
        (
        1
        −
        μ
        
          )
          
            n
            −
            r
          
        
        .
      
    
    {\displaystyle P(X=r)={n \choose r}\mu ^{r}(1-\mu )^{n-r}.}
  

This is valid only if the probability of more than one occurrence per unit time 
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
 is zero. Often that is a close approximation, in which case the probabilities yielded by this formula hold approximately.
If 
  
    
      
        n
        →
        ∞
        ,
        μ
        →
        0
      
    
    {\displaystyle n\rightarrow \infty ,\mu \rightarrow 0}
  
 in such a way that 
  
    
      
        n
        μ
        →
        λ
      
    
    {\displaystyle n\mu \rightarrow \lambda }
  
 then

  
    
      
        
          
            
              n
              !
            
            
              (
              n
              −
              r
              )
              !
              r
              !
            
          
        
        
          μ
          
            r
          
        
        (
        1
        −
        μ
        
          )
          
            n
            −
            r
          
        
        →
        
          e
          
            −
            λ
          
        
        
          
            
              λ
              
                r
              
            
            
              r
              !
            
          
        
        .
      
    
    {\displaystyle {\frac {n!}{(n-r)!r!}}\mu ^{r}(1-\mu )^{n-r}\rightarrow e^{-\lambda }{\frac {\lambda ^{r}}{r!}}.}
  

Take

  
    
      
        μ
        =
        
          
            1
            T
          
        
        =
        
          
            m
            
              n
              +
              1
            
          
        
      
    
    {\displaystyle \mu ={\frac {1}{T}}={m \over n+1}}
  

where

T is return interval
n is number of years on record.
m is the number of recorded occurrences of the event being considered

=== Example ===
Given that the return period of an event is 100 years,

  
    
      
        p
        =
        
          
            1
            100
          
        
        =
        0.01.
      
    
    {\displaystyle p={1 \over 100}=0.01.}
  

So the probability that such an event occurs exactly once in 10 successive years is:

  
    
      
        
          
            
              
                P
                (
                X
                =
                1
                )
              
              
                
                =
                
                  
                    
                      (
                    
                    
                      10
                      1
                    
                    
                      )
                    
                  
                
                ×
                
                  0.01
                  
                    1
                  
                
                ×
                
                  0.99
                  
                    9
                  
                
              
            
            
              
              
                
                ≈
                10
                ×
                0.01
                ×
                0.914
              
            
            
              
              
                
                ≈
                0.0914
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}P(X=1)&={\binom {10}{1}}\times 0.01^{1}\times 0.99^{9}\\[4pt]&\approx 10\times 0.01\times 0.914\\[4pt]&\approx 0.0914\end{aligned}}}
  

=== Risk analysis ===
Return period is useful for risk analysis (such as natural, inherent, or hydrologic risk of failure). When dealing with structure design expectations, the return period is useful in calculating the riskiness of the structure.
The probability of at least one event that exceeds design limits during the expected life of the structure is the complement of the probability that no events occur which exceed design limits.
The equation for assessing this parameter is

  
    
      
        
          
            R
            ¯
          
        
        =
        1
        −
        
          
            (
            
              1
              −
              
                
                  1
                  T
                
              
            
            )
          
          
            n
          
        
        =
        1
        −
        (
        1
        −
        P
        (
        X
        ≥
        
          x
          
            T
          
        
        )
        
          )
          
            n
          
        
      
    
    {\displaystyle {\overline {R}}=1-\left(1-{1 \over T}\right)^{n}=1-(1-P(X\geq x_{T}))^{n}}
  

where

  
    
      
        
          
            1
            T
          
        
        =
        P
        (
        X
        ≥
        
          x
          
            T
          
        
        )
      
    
    {\displaystyle {1 \over T}=P(X\geq x_{T})}
  
 is the expression for the probability of the occurrence of the event in question in a year;
n is the expected life of the structure.

== See also ==
100-year flood
Cumulative frequency analysis
Frequency of exceedance
Residence time

== References ==