Odds and Probability Formula with Solved Examples
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 probability in Maths?
Probability is the measure of how likely an event is to occur, expressed as a number between 0 and 1. In probability theory:
- 0 means the event is impossible.
- 1 means the event is certain.
- It is calculated using the formula: Probability = (Number of favourable outcomes) / (Total number of possible outcomes).
2. What is the formula for calculating probability?
The basic probability formula is P(E) = n(E) / n(S), where E is the event and S is the sample space. Here:
- n(E) = number of favourable outcomes
- n(S) = total number of possible outcomes
3. What are odds in probability?
Odds compare the number of favourable outcomes to the number of unfavourable outcomes. The formula for odds in favour of an event is Odds = Favourable outcomes : Unfavourable outcomes. For example, when rolling a die, the odds in favour of getting a 3 are 1:5 because there is 1 favourable outcome and 5 unfavourable outcomes.
4. What is the difference between odds and probability?
Probability measures likelihood as a fraction of total outcomes, while odds compare favourable outcomes to unfavourable outcomes. Key differences:
- Probability = Favourable / Total outcomes
- Odds = Favourable : Unfavourable
5. How do you calculate the probability of an event step by step?
To calculate probability, divide favourable outcomes by total possible outcomes. Follow these steps:
- Step 1: Identify the sample space (S).
- Step 2: Count the favourable outcomes (E).
- Step 3: Apply P(E) = n(E)/n(S).
6. What is the probability of two independent events?
The probability of two independent events occurring together is the product of their individual probabilities, given by P(A ∩ B) = P(A) × P(B). Independent events do not affect each other. For example, probability of getting two heads when tossing a coin twice is (1/2) × (1/2) = 1/4.
7. What is conditional probability?
Conditional probability is the probability of an event occurring given that another event has already occurred, calculated using P(A|B) = P(A ∩ B) / P(B). It measures how one event affects another. For example, the probability of drawing a king given that the card drawn is a face card depends only on the 12 face cards in the deck.
8. What is a sample space in probability?
A sample space is the set of all possible outcomes of a random experiment. It is usually denoted by S. For example:
- Coin toss: S = {H, T}
- Die roll: S = {1, 2, 3, 4, 5, 6}
9. Can probability be greater than 1?
No, probability cannot be greater than 1 because it represents a value between 0 and 1 inclusive. A value greater than 1 would mean the event is more than certain, which is mathematically impossible. The rule is: 0 ≤ P(E) ≤ 1.
10. What are some real-life examples of odds and probability?
Odds and probability are used to measure uncertainty in real-life situations. Common examples include:
- Weather forecasting (probability of rain)
- Games and gambling (odds in betting, dice, cards)
- Medical testing (probability of disease)
- Insurance risk assessment
