Odds in Favour or Against an Event

Odds in Favour or Against an Event:

Odds in Favour of an Event- It is defined as the ratio of the number of favorable cases of an event to the number of unfavorable cases (i.e.)

odds in favor of an event A = Number of favorable cases/Number of unfavorable cases

Odds Against an Event- It is defined as the ratio of the number of unfavorable cases of an event to the number of favorable cases of an event i.e.

odds against an event A = Number of unfavorable cases/Number of favorable cases

If m be the number of cases favorable to an event A and n be the unfavorable cases.

∴ odds in favor of A = m/n
⇒ P(A) = m/(m+n)
and P(Ā) = n/(m+n)

Example 1- The odds in favor of an event are 3:4. Find the probability of- (i) occurrence of an event (ii) non-occurrence of an event.

Solution- odds in favor = 3/4
∴ number of favorable cases = 3x
and number of unfavorable cases = 4x

∴ Total number of exhaustive cases = 3x + 4x = 7x

(i) P(occurrence of an event) = 3x/7x = 3/7
(ii) P(non-occurrence of an event) = 4x/7x = 4/7

Example 2- The probability of an occurrence of an event is 3:5. Find the odds against this event.

Solution- Let A be the event such that
P(A) = 3/5

⇒ P(Ā) = 1 – P(A)
⇒ P(Ā) = 1 -3/5
⇒ P(Ā) = 2/5

∴ odds against an event = P(Ā)/P(A)
⇒ odds against an event = (2/5)/(3/5)
⇒ odds against an event = 2/3