Probability - Aptitude Tricks, Formulas & Shortcuts

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This is an important topic for all of those students appearing in competitive exams like Placement test, Bank PO, CAT, XAT, SNAP etc. because the questions from probability will test your counting skills such as permutation and combination too. Once you are really skilled in combinatorics (Basic counting methods, permutation and combination) you can easily solve the aptitude questions from this area by using common sense. In this article, we provide - basic terminologies of probability, important formulas, basic and advanced concepts, Baye's Theorem, probability aptitude question solving tricks.

The chance to occur an event is a simple and basic definition for the term probability.
So, probability of an event = No of favourite outcomes of the event / Total number of possible outcomes


Probability : Basic Terminologies, Formulas & Concepts

1. Deterministic and probabilistic (or Random experiments)
If the outcome of an operation is certain, then the operation is called a deterministic experiment, then there is no scope of probability.
E.g. Direction of raising the sun is a deterministic natural phenomenon. Most of the Laboratory experiment are deterministic like simple pendulum, glass prism etc.
If the outcome of any operation is not predictable, then the operation is not deterministic, then it is called a probabilistic (or random) experiment.

E.g. winning of a team in a cricket match, Getting rain in a particular place etc.

2. Random Experiment:

If an experiment is not deterministic and it gives any of the possible outcomes, then the experiment is called a random experiment.

E.g. tossing of a fair coin, Rolling of an unbiased die.

3. Sample Space:

The set of all possible outcomes of a random experiment is called sample space.

E.g. When rolling n unbiased die, then there are 6 possible outcomes as the result.The set of all possible outcomes can be arranged in a set in the form S={1,2,3,4,5,6}

This set is the sample space of the event of rolling a die.

4. Event

Any subset of a sample space is called an event.

e.g. In a flipping of coin ,the event of getting a head can be considered as {H}.This event is a subset of the sample space{H,T} of the experiment.

5. Elementary Event

Basically it is a singleton subset of the sample space of any operation. An event containing only a particular outcome of an operation is called an elementary event.

E.g. {1},{2},{3},{4},{5},{6} are the six different elementary events of the operation "rolling an unbiased die".

6. Compound Events

If an event contains more than one element from the sample space of an operation is called a compound event.

E.g. an event of getting even numbers in the rolling of a die is {2, 4, 6} and this is a compound event.

7. Impossible Event

Basically it is an empty event or it is the not happening condition (i.e. it is an event which can never occur under any circumstances).

E.g. Event of getting a two digit number in the rolling of die is {} or

8. Sure Event

It is an event which is sure to occur. It is representing the sample space itself.

E.g. the event of getting a single digit number in throwing a die is a sure event.

9. Equally Likely Event

If there is not any preference for any of the possible outcomes in an event (experiment) or the chance of occurrence of the outcome is equally distributed in all the possible outcomes in an event is called an equally likely event.

E.g. while flopping a fair coin, the chance to get a head or tail is equally distributed.

10. Complementary Events

An event is representing the complementary of any other event E is called the complementary event and it is denoted by Ec

E U Ec = sample space

E.g. Let E be the event of getting an odd number in through a die and the Ec is the event of not getting an odd number in same operation.

11. Mutually Exclusive Events

If two events E1 and E2 are totally different, then E1 and E2 are called mutually exclusive events. Or in another words, one of the events always prevents the happening of the other.

E1 ∩ E2 = ψ

E.g. consider the operation of flipping a coin .Let E1 be the event of getting a head and E2 be the event of getting a tail.

E1= {H}
E2= {T}
E1 ∩ E2 = ψ

Consider another example of drawing a card from a well shuffled pack of playing cards.

E1 - An event of getting a heart.

E2 - An event of getting a spade.

Here E1c is the complementary of E1 but E2 is not a complementary of E1 because

E1 ∪ Ec = sample space, but E1 ∪ E2 ≠ sample space

E1 and E2 are mutually exclusive events.

i.e. E1 ∩ E2 = ∅

12. Probability of an event

Let E be an event and Ec is its complementary event. If S is the sample space, then the probability of an event E,

p(E) = n(E) / n(S)

p(Ec) = n(Ec) / n(S)

= n(S) - n(E) / n(S)

= 1 - (n(S) / n(E))

= 1 - p(E) ie. p(Ec) = 1 - p(E)


p(E) + p(Ec) = 1

Probability of an event E normally represented by 0, proper fraction or 1.

i.e. 0 ≤ p(E) ≤ 1

If p (E) = 0 then the event is an impossible event.

If p (E) = 1 then the event is a certain event.

In general, for any event E;

0 ≤ n(E) ≤n(S)

0 ≤ n(E) / n(S) ≤ 1

0 ≤ p(E) ≤ 1

13. Odds in favor of an event

It is basically a ratio of the probability of an event E and the complementary event Ec.

i.e. odds in favor of an event E = p(E) / p(Ec)

E.g. Odds in favor of getting prime number in rolling of a die is

n({2,3,5}) / n({1,4,6}) = 3 / 3 = 1 V 1 : 1

14. Odds against an event

It is the ratio of the probabilities of the complementary of an event to the event itself.

i.e. odds against an event E is p(Ec) / p(E)

E.g. Let E be the event of getting a multiple of 3 in rolling a die.

Then E = {3,6} and Ec = {1,2,4,5}

Odds against the event,

E = p(Ec) / p(E)

= n(Ec) / n(E)

= 4 / 2

= 2:1
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