# Odds in Favor of an Event

Odds in Favor of an Event is the ratio of Number of Favorable Choices or Successes for the event to the Number of Unfavourable Choices or Failures for the event.

⇒ Odds in Favor of an Event

 = Number of Favorable Choices : Number of Unfavorable Choices Or Number of Successes : Number of Failures = m : mc Or = m : (n − m)

# Odds against an Event

Odds against an Event is the ratio of Number of Unfavorable Choices or Failures for the event to the Number of Favorable Choices or Successes for the event.

⇒ Odds against an Event

 = Number of Unfavourable Choices : Number of Favorable Choices Or Number of Failures : Number of Successes = mc : m Or = (n − m) : m

# Finding Odds using Probability

Probabilities for and against the event can be used as the antecedent and consequent of the ratio representing the odds for an event in place of favorable and unfavorable choices.

## 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 : mc
=
 m m + mc
:
 mc m + mc
=
 m n
:
 mc n
=

P(Event) : P(Eventc)

Or = m : (n − m)
=
 m m + (n − m)
:
 n − m m + (n − m)
=
 m n
:
 n − m n
= P(Event) : P(Eventc)

⇒ Odds in Favor of an Event = P(Event) : P(Eventc)

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

= mc : m
=
 mc mc + m
:
 m mc + m
=
 mc n
:
 m n
=

P(Eventc) : P(Event)

Or = (n − m) : m
=
 n − m (n − m) + m
:
 m (n − m) + m
=
 n − m n
:
 m n
= P(Eventc) : P(Event)

⇒ Odds against an Event = P(Eventc) : P(Event)

# Finding Probability using Odds in Favor

Let Odds in Favor of the Event be x : y.

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 : mc

x : y = m : mc

If k is the common factor between m and mc,

• m = kx and
• mc = ky

Total number of possible choices

 = Number of Favorable Choices + Number of Unfavorable Choices Or Number of Successes + Number of Failures
 n = m + mc = kx + ky = k (x + y)

Probability of Occurrence of the Event

Or

Probability of Success for the Event

=
 Number of Favorable Choices or Successes for the Event Total Number of Possible Choices for the Experiment
⇒ P(E) =
 m n
=
 kx k(x + y)
=
 x x + y
=
 antecedent antecedent + consequent

Probability of Non Occurrence of the Event

Or

Probability of Failure for the Event

=
 Number of Unfavorable Choices or Failures for the Event Total Number of Possible Choices for the Experiment
⇒ P(Ec) =
 mc n
=
 ky k(x + y)
=
 y x + y
=
 consequent antecedent + consequent

Where, odds in favor of an event is x : y,

P(Event) =
 x x + y
P(Eventc) =
 y x + y

# Finding Probability using Odds against

Let Odds in Favor of the Event be p : q.

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 = mc : m

p : q = mc : m

If a is the common factor between mc and m,

• mc = 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 + mc = qa + pa = a (q + p) = a (p + q)

Probability of Occurrence of the Event

Or

Probability of Success for the Event

=
 Number of Favorable Choices or Successes for the Event Total Number of Possible Choices for the Experiment
⇒ P(E) =
 m n
=
 qa a(p + q)
=
 q p + q
=
 consequent antecedent + consequent

Probability of Non Occurrence of the Event

Or

Probability of Failure for the Event

=
 Number of Unfavorable Choices or Failures for the Event Total Number of Possible Choices for the Experiment
⇒ P(Ec) =
 mc n
=
 pa a(p + q)
=
 p p + q
=
 antecedent antecedent + consequent

Where, odds against an event is p : q,

P(Event) =
 q p + q
P(Eventc) =
 p p + q