Probability: Concepts & Tricks

 Introduction to Probability

Just like permutation and combination, the probability word problems appear frequently in the competitive exams. In this article, we will cover basic probability theory and probability equations.

A classic definition of probability is the chance or the likelihood that an event will occur, which means the ratio of the number of favorable cases to the total number of possible cases, provided that all cases are equally likely. Probability always lies between 0 and 1.

If the probability of happening of an event is 0, then it is an impossible event.

If the probability of happening of an event is 1, then it is a  sure event.

Probability of happening of any event P(A) = fav. number of cases / Total no. of cases = n/N

Now while solving probability in math, we have to use some specific definitions of the topic, which are given below.

While studying probability theory, we will frequently use the term ‘experiment' which means an operation which can produce well defined outcome(s). There are two types of experiments:

(i) Deterministic Experiment: The experiments whose outcome is same when done under exact conditions are called Deterministic experiment. E.g. all experiments are done in chemistry lab.

(ii) Random Experiment: The experiments whose outcomes are more than 1 when done under exact conditions are called Random Experiment. E.g. if a coin is tossed we may get a head or a tail.

When we perform any experiment, there are some outcomes which are called events. Let us study the different types of events can occur.

Trial and Elementary Events: If we repeat a random experiment under exact conditions, it is known as trial and all the possible outcomes are known as elementary events. E.g. if we throw a dice it is called a trial and getting 1, 2, 3, 4, 5 or 6 is called elementary event.

Compound Event: When two or more elementary events are combined it is known as compound event. When we throw a dice, getting a prime number is compound event as we can get 2, 3, 5 and all are elementary.

Exhaustive Number of Cases: It is the total possible outcome. When we throw a dice total number of cases are 6. When we throw a pair of dice exhaustive number of cases is 36.

Mutually Exclusive Events: It means simultaneous occurrence is not possible. In case of tossing a coin, either head will come or tail will come. So, both are mutually exclusive events.

Equally Likely Cases: It means chances are equal. When we throw a dice, each outcome has equal chance. So it is case of equally likely.

Total Number of Cases: As the name suggests, the total number of elementary events of a trial are known as total number of cases.

Favorable Events: Desired outcome of an elementary event is called Favorable event. E.g. when we throw a dice and it is asked that what is the probability of getting a multiple of 3? In this case favorable cases are 2 (3 and 6) and total cases are obviously 6.

Independent Events: Two events are called independent if outcome of one event is not affecting the outcome of other. If we toss a coin and throw a dice then outcome of coin is independent of outcome of coin, both are independent events.

Probability in simple language is defined as ratio of favorable cases to the total number of cases.

Probability of happening of any event P(A) = fav. number of cases / Total no. of cases = n/N

If p is the probability of happening of an event A, then  the probability of not happening of that event is P(Ā) = 1- p

Probability Equations: P (A) ≤ 1, P(A) + P(Ā) = 1.

Addition theorem: P(X or Y) = P(X) + P(Y) – P (X∩Y)

or P(X⋃Y) = P(X) + P(Y) – P(X∩Y)

Mutually exclusive events: Two events are mutually exclusive if they cannot occur simultaneously. For n mutually exclusive events, the probability is the sum of all probabilities of these events:

p = p1 + p2 + ... + p (n-1) + p (n)

or

P (A or B) = P (A) + P (B) where A and B denote mutually exclusive events. 

Independent events: Two events are independent if the occurrence of one event does not influence the occurrence of other events. Therefore, for n independent events, the probability is the product of all probabilities of independent events:

p = p1 x p2 x ... x p (n-1) x p (n)

or P(X and Y) = P(X) x P(Y) , where X and Y denote independent events

For solving the questions on probability, you are advised to revise the major probability formulas, go through 20 to 25 probability examples & solutions and solve around 100 probability sums. After doing this, you will feel confident to solve probability problems on your own.