Odds in Favor and Odds against an Event. Calculating  Odds from Probabilities, Probabilities from Odds
Odds in Favor of an Event
⇒ Odds in Favor of an Event
=  Number of Favorable Choices : Number of Unfavorable Choices Or Number of Successes : Number of Failures  
=  m : m^{c}  
Or  =  m : (n − m) 
Odds against an Event
⇒ Odds against an Event
=  Number of Unfavourable Choices : Number of Favorable Choices Or Number of Failures : Number of Successes  
=  m^{c} : m  
Or  =  (n − m) : m 
Finding Odds using Probability
Odds in Favor of an Event
Odds in Favor of an Event
=  Number of Favorable Choices : Number of Unfavorable Choices Or Number of Successes : Number of Failures  
=  m : m^{c}  
= 
 
= 
 
=  P(Event) : P(Event^{c})  
Or  =  m : (n − m)  
= 
 
= 
 
=  P(Event) : P(Event^{c}) 
⇒ Odds in Favor of an Event = P(Event) : P(Event^{c})
Probabilities against and for the event can be used as the antecedent and consequent of the ratio representing the odds against an event in place of unfavorable and favorable choices.
Odds against an Event
Odds against an Event
=  Number of Unfavourable Choices : Number of Favorable Choices Or Number of Failures : Number of Successes  
=  m^{c} : m  
= 
 
= 
 
=  P(Event^{c}) : P(Event)  
Or  =  (n − m) : m  
= 
 
= 
 
=  P(Event^{c}) : P(Event) 
⇒ Odds against an Event = P(Event^{c}) : P(Event)
Finding Probability using Odds in Favor
For the ratio representing odds in favor
antecedent = x and consequent = y
Odds in Favor of an Event
=  Number of Favorable Choices : Number of Unfavorable Choices Or Number of Successes : Number of Failures  
=  m : m^{c} 
x : y = m : m^{c}
If k is the common factor between m and m^{c},
 m = kx and
 m^{c} = ky
Total number of possible choices
=  Number of Favorable Choices + Number of Unfavorable Choices Or Number of Successes + Number of Failures 
n  =  m + m^{c} 
=  kx + ky  
=  k (x + y) 
Probability of Occurrence of the Event
OrProbability of Success for the Event
= 

⇒ P(E)  = 
 
= 
 
= 
 
= 

Probability of Non Occurrence of the Event
OrProbability of Failure for the Event
= 

⇒ P(E^{c})  = 
 
= 
 
= 
 
= 

Where, odds in favor of an event is x : y,
x 
x + y 
y 
x + y 
Finding Probability using Odds against
For the ratio representing odds in favor
antecedent = p and consequent = q
Odds against an Event
=  Number of Unfavourable Choices : Number of Favorable Choices Or Number of Failures : Number of Successes  
=  m^{c} : m 
p : q = m^{c} : m
If a is the common factor between m^{c} and m,
 m^{c} = pa and
 m = qa
Total number of possible choices
=  Number of Favorable Choices + Number of Unfavorable Choices Or Number of Successes + Number of Failures 
n  =  m + m^{c} 
=  qa + pa  
=  a (q + p)  
=  a (p + q) 
Probability of Occurrence of the Event
OrProbability of Success for the Event
= 

⇒ P(E)  = 
 
= 
 
= 
 
= 

Probability of Non Occurrence of the Event
OrProbability of Failure for the Event
= 

⇒ P(E^{c})  = 
 
= 
 
= 
 
= 

Where, odds against an event is p : q,
q 
p + q 
p 
p + q 