The word probably means likely or chance. We use probability to denote the happening of a certain event, and the occurrence of that event, based on past experiences. We use the word odd frequently in statistics. Let us now discuss odds in probability in detail.
Odds
Before we study odds we need to understand some useful terms.
Some Useful Terms
- Random Experiment or Trial: It refers to an action or an operation which can produce any result or outcome.
- Event: Event is any possible outcome of a random experiment.
- Sample space: The set of all possible outcomes of an experiment is called sample space. For example, when we roll a die, the possible outcomes are 1, 2, 3, 4, 5, and 6.
- Mutually exclusive events or cases: The occurrence of which prevents the possibility of the other to occur. For example, if a coin is tossed, either the head can be up or tail can be up, but both cannot be up at the same time.
- Equally likely events: Two or more events are said to be equally likely if the chance of their happening is equal. For example, the head and tail are equally likely events in tossing an unbiased coin.
- Exhaustive events: Exhaustive events refer to the total number of possible outcomes of a random experiment. For example, tossing a coin, the possible outcome is head or tail, exhaustive events are two.
- Independent Events: A set of events is said to be independent if the occurrence of an outcome of them does not affect the occurrence of any other outcome in the set.
- Dependent Events: Dependent events are those the occurrence of one event affects the happening of the other events.
- Favorable Cases: The number of outcomes which result in the happening of the desired event is called favorable cases to the event. For example, in a single throw of dice, the number of favorable cases of getting an odd number are three -1, 3 and 5.
Odds in Probability
For measuring probabilities, the probability is the ratio of favorable events to the total number of equally likely events.
P = \(\frac{Number of favorable cases}{Total number of cases}\)
Therefore, P + q = 1
Where P is the probability of the event happening and q is the probability of it’s not happening.
Odds relate the chances in favour of an event to the chances against it. For instance, the odds are 2: 1 that A will get a job, means that there are 2 chances that he will get the job and 1 chance against his getting the job. This can also be converted into probability as getting the job = \(\frac{2}{3}\).
Therefore, if the odds are a : b in favour of an event then P (A) = \(\frac{a}{a+b}\). If the probability of an event is p, then the odds in favor of its occurrence are P to (1-p) and the odds against its occurrence are 1-p to p.
Solved Example on Odds
Q. There are 12 to 7 against a person A who is now 40 years of age living till he is 73 and 8 to 5 against B who is 45 living till he is 78. Find the chance that at least one of these persons will be alive 33 years from now.
Solution :
The probability that A will die within 33 years = \(\frac{12}{19}\)
The probability that B will die within 33 years = \(\frac{8}{13}\)
Probability that both of them will die within 33 years = \(\frac{12}{19}\) X \(\frac{8}{13}\) = \(\frac{96}{247}\)
The probability that atleast one of them will be alive = 1- \(\frac{96}{247}\)
= \(\frac{151}{247}\)
In percentage
\(\frac{151}{247}\) X 100 = 61.1%
There are so many errors in two of the lectures that I have watched. The flow of the lectures are also inappropriate. Firstly you never defined what an event is. For this lecture you can just say that an event is a subset of sample space. Therefore it can be any subset of sample space, even phi(empty set) or the whole sample space itself. You are confusing events with elements of sample space.
There is a fundamental errors on tis page too. Like P(A|B) is probability of event A given that event B has (already) occurred. However the text in this page says the other way.
Please revise it before making public.
Your well-wisher
Kumar.
@Kumar Thank you! I thought I was somehow wrong in my understanding. Another error is: “Even compound events (two events occurring at the same time) can be independent events.
Ex. Tossing a coin and rolling a die.
Sample space S = {(1,H), (2, H), (3, H), (4, H), (5, H), (6, H), (1, T), (2, T), (3, T), (4, T) (5, T) (6, T)}.”
The example is still simple even and not a compound event. An actual example for a compound event will be ex: Tossing a coin and rolling a die and getting a 1 every time. The sample space is S as given but the even space is E= {(1,H),(1,T)} this is a compound event as the event has 2 possible outcomes that fit and not just one. ( and also they are independent.)
@Kumar Thank you! I thought I was somehow wrong in my understanding. Another error is: “Even compound events (two events occurring at the same time) can be independent events.
Ex. Tossing a coin and rolling a die.
Sample space S = {(1,H), (2, H), (3, H), (4, H), (5, H), (6, H), (1, T), (2, T), (3, T), (4, T) (5, T) (6, T)}.”
The example is still simple even and not a compound event. An actual example for a compound event will be ex: Tossing a coin and rolling a die and getting a 1 every time. The sample space is S as given but the even space is E= {(1,H),(1,T)} this is a compound event as the event has 2 possible outcomes that fit and not just one. ( and also they are independent.)