If the odds in favour of an event are given as p:q, what is the probability that the event will occur?

Introduction / Context:
This conceptual question links the idea of odds with the usual definition of probability. In many competitive exams and real life situations, information is sometimes expressed as odds in favour of an event rather than as a simple probability. Understanding how to convert between odds and probability is therefore an important skill.

Given Data / Assumptions:

• The odds in favour of a certain event are given as p:q.
• This means that for every p favourable cases, there are q unfavourable cases.
• All cases under consideration are assumed to be equally likely.
• We want the probability that the event occurs, expressed in terms of p and q.

Concept / Approach:
Odds in favour of an event describe the ratio of favourable outcomes to unfavourable outcomes. If the odds are p:q, then there are p favourable outcomes and q unfavourable outcomes. The total number of possible outcomes is p plus q. Probability, by definition, is the ratio of favourable outcomes to total outcomes. So we simply divide the number of favourable cases by the sum of favourable and unfavourable cases.

Step-by-Step Solution:
Odds in favour p:q means favourable outcomes = p and unfavourable outcomes = q. Total outcomes under consideration = p + q. Probability of the event occurring = favourable outcomes / total outcomes. So probability = p / (p + q).

Verification / Alternative check:
To verify, consider a simple numerical example. Suppose the odds in favour of an event are 3:2. This implies 3 favourable and 2 unfavourable cases, giving total outcomes equal to 5. The probability of the event is then 3 / 5, which matches the formula p / (p + q) when p equals 3 and q equals 2. Trying other values, such as p equal to 1 and q equal to 4, gives probability 1 / 5, which also fits the formula.

Why Other Options Are Wrong:
p/q represents the odds ratio itself, not the probability. It compares favourable to unfavourable cases but does not include the total number of outcomes. q/(p+q) actually gives the probability that the event does not occur, because it uses the count of unfavourable outcomes over the total. q/p is simply the odds against the event, which is the reciprocal of the odds in favour, not the required probability.

Common Pitfalls:
Learners sometimes confuse odds and probability and attempt to use p/q directly as a probability. This leads to values greater than 1 when p is larger than q, which cannot be a valid probability. Always remember that probability must lie between 0 and 1 and that odds must be converted by dividing the favourable part by the total of favourable plus unfavourable cases.

Final Answer:
If the odds in favour of an event are p:q, then the probability that the event occurs is p/(p+q).