How to Solve Odds and Probability Questions Step by Step
When we do something, in mathematics we call it an event. So when an event occurred then there is some outcome or result of that event, to study that outcome or get an idea of an event we use three-term which are odds, chances, and probability
Odds and Probability in Mathematics
Definition of Odds
Odds:- It is a measure of the likelihood of a particular outcome and it is generally calculated by the ratio of the number of favorable outcomes to the number of unfavorable outcomes
i.e
\[{\rm{Odds = }}\dfrac{{{\rm{Number \; of \; favourable\; outcomes}}}}{{{\rm{Number \; of \; total \; outcomes}}}}\]
Definition of Probability and Chances
Probability:- It is the chance that something will happen, or how likely it is that an event will occur. It is calculated by the ratio of the number of favorable outcomes to the number of total outcomes
i.e
\[{\rm{probability = }}\dfrac{{{\rm{Number\; of \; favourable\; outcomes}}}}{{{\rm{Number \; of \; total \; outcomes}}}}\]
Calculating the Probability of an event
\[{\rm{Probability \; of \; event = }}\dfrac{{{\rm{No}}{\rm{. of \; favourable\; outcomes}}}}{{{\rm{No}}{\rm{. of \; favourable\; outcomes + No}}{\rm{. of\; unfavourable \; outcomes}}}}\]
Chances:- It is, simply, the possibility of something happening, which is not planned or controlled. Its value is the same as probability.
Odds in Favour
Odds in favour of a particular event are given by the Number of favorable outcomes to the Number of unfavorable outcomes.
i.e Odds Formula
\[{\rm{odds\; in\; favour = }}\dfrac{{{\rm{No}}{\rm{. of favourable\; outcomes}}}}{{{\rm{No}}{\rm{. \;of \;unfavourable\; outcomes}}}}\]
Odds in Against
Odds against are given by Number of unfavorable outcomes to the number of favorable outcomes.
i.e
\[{\rm{Odds\; in\; Against = }}\dfrac{{{\rm{No}}{\rm{. of \;unfavourable \;outcomes}}}}{{{\rm{No}}{\rm{. of\; favourable\; outcomes}}}}\]
Difference between Odds and Probability
The difference between odds and probability are:
Odds of an event are the ratio of success to failure.
\[{\rm{Odds = }}\dfrac{{{\rm{success}}}}{{{\rm{failure}}}}\]
The probability of an event is the ratio of success to the sum of success and failure.
\[{\rm{Probability = }}\dfrac{{{\rm{success}}}}{{{\rm{success + failure}}}}\]
Solved Examples
1. Find the odds in favor of throwing a die to get “3 dots”.
Solution:
Total number of outcomes in throwing a die = 6 (1,2,3,4,5,6)
Number of favorable outcomes = 1 (3)
Number of unfavorable outcomes = (6 - 1) = 5
Therefore, odds in favor of throwing a die to get “3 dots” is 1 : 5 or \[\dfrac{1}{5}\]
2. Find the odds in favor of throwing a coin to get a “tail”.
Solution:
Total number of outcomes in throwing a coin = 2 (“head”,”tail”)
Number of favorable outcomes = 1 (“tail”)
Number of unfavorable outcomes = (2- 1) = 2
Therefore, odds in favor of throwing a coin to get a “tail” is 1 : 1 or \[\dfrac{1}{1}\]
3. Find the odds against throwing a die to get “3 dots”.
Solution:
Total number of outcomes in throwing a die = 6
Number of favorable outcomes = 1
Number of unfavorable outcomes = (6 - 1) = 5
Therefore, odds in against of throwing a die to get “3 dots” is 5 : 1 or \[\dfrac{5}{1}\]
4. Find the odds against throwing a die to get “2 dots”.
Solution:
Total number of outcomes in throwing a die = 6(1,2,3,4,5,6)
Number of favorable outcomes = 1
Number of unfavorable outcomes = (6 - 1) = 5
Therefore, odds in against of throwing a die to get “3 dots” is 5 : 1 or \[\dfrac{5}{1}\]
5.Find the probability of getting “2 dots” in throwing a die.
Solution:
Total number of outcomes in throwing a die = 6
Number of favorable outcomes = 1
Number of unfavorable outcomes = (6 - 1) = 5
Therefore, probability of getting “2 dots” in throwing a die.
is 1: (1+5) or \[\dfrac{1}{6}\]
6.If odds in favor of X solving a problem are 4 to 3 and odds against Y solving the same problem are 2 to 6.
Find probability for:
(i) X solving the problem
(ii) Y solving the problem
Solution:
Given odds in favor of X solving a problem are 4 to 3.
Number of favorable outcomes = 4
Number of unfavorable outcomes = 3
(i) X solving the problem
P(X) = P(solving the problem) = 4/(4 + 3)
= \[\dfrac{4}{7}\]
Given odds against Y solving the problem are 2 to 6
Number of favorable outcomes = 6
Number of unfavorable outcomes = 2
(ii) Y solving the problem
P(Y) = P(solving the problem) = 6/(2 + 6)
= \[\dfrac{6}{8}\]
= \[\dfrac{3}{4}\]
In this article we learned about Odds and probability and how to calculate odds and also learnt the physical meaning of both
FAQs on Odds and Probability Explained for Students
1. What is the fundamental difference between odds and probability?
The main difference lies in what is being compared. Probability measures the likelihood of an event by comparing the number of favourable outcomes to the total number of possible outcomes. In contrast, odds measure likelihood by comparing the number of favourable outcomes to the number of unfavourable outcomes.
2. How do you calculate the 'odds in favour' of an event?
To calculate the 'odds in favour' of an event, you use the following ratio:
Odds in Favour = (Number of favourable outcomes) / (Number of unfavourable outcomes).
For example, when rolling a standard six-sided die, the odds in favour of getting a '4' are 1 to 5, because there is one way to succeed (rolling a 4) and five ways to fail (rolling a 1, 2, 3, 5, or 6).
3. How are 'odds against' an event different from 'odds in favour'?
‘Odds against’ an event is simply the reciprocal of ‘odds in favour’. It compares the number of unfavourable outcomes to the number of favourable outcomes. The formula is:
Odds Against = (Number of unfavourable outcomes) / (Number of favourable outcomes).
Using the die-roll example, the odds against rolling a '4' are 5 to 1.
4. How can you convert a given probability into odds?
You can convert probability (P) to odds using a simple formula. If 'P' is the probability of an event occurring, then (1-P) is the probability of it not occurring. The odds in favour are the ratio of these two probabilities.
- Formula: Odds in Favour = P / (1 - P)
- Example: If the probability of winning a game is 25% (or 0.25), the odds in favour of winning are 0.25 / (1 - 0.25) = 0.25 / 0.75 = 1/3. This is expressed as odds of 1 to 3.
5. Can the odds of an event be greater than 1? Explain why.
Yes, the odds of an event can certainly be greater than 1. This happens whenever an event is more likely to occur than not to occur. Since odds are a ratio of favourable to unfavourable outcomes, if there are more ways to succeed than to fail, the numerator will be larger than the denominator. For example, the odds of drawing a red card from a standard deck are 26 to 26 (or 1), but the odds of drawing a non-ace card are 48 to 4, which is 12, a value much greater than 1.
6. Why are odds and probability expressed differently if they both measure likelihood?
They are expressed differently because they answer slightly different questions and provide different perspectives on risk and chance. Probability gives a sense of proportion and tells you what fraction of the time you can expect a certain outcome out of all possibilities. Odds provide a direct comparison between success and failure, which can be more intuitive for understanding the balance of risk, such as in games or betting scenarios. It frames the situation as 'chances for' versus 'chances against'.
7. In a real-world scenario, when is it more useful to think in terms of odds instead of probability?
Thinking in terms of odds is often more useful in situations involving risk assessment, betting, and strategic decision-making. For instance, in sports betting or horse racing, odds directly relate to the payout structure for a wager. A bookmaker might set odds of 5 to 1 against a team winning, which directly tells a gambler their potential return. In these contexts, the direct comparison of failure to success (risk vs. reward) is more actionable than the general probability of success.
