Probability: Introduction and Definition | Mathematics (Maths) for JEE Main & Advanced PDF Download

Introduction:

The theory of probability originated from the game of chance and gambling. In days of old, gamblers used to gamble in a gambling house with a die to win the amount fixed among themselves. They were always desirous to get the prescribed number on the upper face of a die when it was thrown on a board. Shakuni of Mahabharat was perhaps one of them. People started to study the subject of probability from the middle of seventeenth century. The mathematicians Huygens, Pascal Fermat and Bernoulli contributed a lot to this branch of Mathematics. A.N. Kolmogorow proposed the set theoretic model to the theory of probability.

We study the occurrence of events which are equally likely in probability. Applying the laws of probability we obtain a formula  in radio activity, where w_t is the probability of an atom not to disintegrate in time t, N_t is the number of atoms that have not disintegrated in time t and N₀ is the number of atoms involved and l is the characteristics constant of the radioactive element. The mathematician Boltzmann in his theorem in classical statistics related entropy to probability. The application of probability theory in quantum mechanics is of greater importance. Here the probability of finding an electron in a state with certain quantum numbers in an element of given volume in the vicinity of a point is determined. This branch of mathematics has applications in applied sciences and statistics also when we plan and organize production. The methods of this theory help in solving the diverse problems of natural science as a result of which its study becomes essential now.

Probability gives us a measure for likelihood that something will happen. However it must be appreciated that probability can never predict the number of times that on occurrence actually happens. But being able to quantify the likely occurrence of an event is important because most of the decisions that affect our daily lives are based on likelihoods and not on absolute certainties.

Definitions:

(a) Experiment : An action or operation resulting in two or more outcome.

(b)Sample space : A set S that consists of all possible outcomes of a random experiment is called a sample space and each outcome is called a sample point often there will be more than one sample space that can describes outcomes of an experiment, but there is usually only one that will provide the most information. If a sample space has a finite number of points it is called finite sample space and infinite sample space if it has infinite number of points.

(c) Event : An event is defined occurrence or situation for example
(i) tossing a coin and the coin landing up head.
(ii) scoring a six on the throw of a die,
(iii) winning the first prize in a raffle.
(iv) being felt a hand of four cards which are all clubs.

In every case it is set of some or all possible outcomes of the experiment. Therefore event (A) is subset of sample space (S). If outcome of an experiment is an element of A we say that event A has occurred.

• An event consisting of a single point of S is called a simple or elementary event.
• If an event has more than one simple point it is called compound event.
• φ is called impossible event and S (sample space) is called sure event.

(d)Complement of an event : The set of all out comes which are in S but not in A is called the Complement Of The Event A denoted by  ,A^c, A’ or ‘not A’.. 

(e)Compound Event : If A and B are two given events then A  ∩ B is called Compound Event and is denoted by A ∩ B or AB or A and B.

(f) Mutually Exclusive Events : Two events are said to be Mutually Exclusive (or disjoint or incompatible) if the occurrence of one precludes (rules out) the simultaneous occurrence of the other. If A and B are two mutually exclusive events then P(A and B) = 0.

Consider, for example, choosing numbers at random from the set {3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If, Event A is the selection of a prime number, Event B is the selection of an odd number, Event C is the selection of an even number, then A and C are mutually exclusive as none of the numbers in this set is both prime and even.
But A and B are not mutually exclusive as some numbers are both prime and odd (viz. 3, 5, 7, 11).

(g) Equally Likely Events : Events are said to be Equally Likely when each event is as likely to occur as any other event.

(h) Exhaustive Events : Events A, B, C..........L are said to be Exhaustive Event if no event outside this set can result as an outcome of an experiment. For example, if A and B are two events defined on a sample space S, then A and B are exhaustive ⇒ A ∪ B = S ⇒ P (A ∪ B) = 1.

Classical Definition of Probability:

Note : (i) 0 ≤ P(A) ≤ 1
(ii) P(A) + P(  ) = 1, Where  = Not A.
(iii) If x cases are favourable to A and y cases are favourable to then P(A) =  and P(  ) = We say that Odds In Favour Of A are x : y and odds against A are y : x.

Comparative study of Equally likely, Mutually Exclusive and Exhaustive events :

Example: Words are formed with the letters of the word PEACE. Find the probability that 2E’s come together
 Sol. Total number of words which can be formed with the letters P, E, A, C, E =  = 60.
Number of words in which 2 E’s come together =

∴ reqd. prob. =

Example:    A bag contains 5 red and 4 green balls. Four balls are drawn at random then find the probability that two balls are of red and two balls are of green colour.
 Sol.  n(s) = the total number of ways of drawing 4 balls out of total 9 balls : ⁹C₄
    A : Drawing 2 red and 2 green balls n(A) = ⁵C₂ x ⁴C₂

Example: If the letters of INTERMEDIATE  are arranged, then the probability no two E’s occur together is
 Sol.    I → 2, N → 1, T → 2, E → 3, R → 1, M → 1, D → 1, A → 1    (3E’s Rest 9)
First arrange rest of letters =
Now 3E’s can be placed into place in ¹⁰C₃ ways so favourable cases =   x ¹⁰C₃ = 3 x 10!
Total cases =  . Probability =  

Example: From a group of 10 persons consisting of 5 lawyers, 3 doctors and 2 engineers, four persons are selected at random. The probability that the selection contains at least one of each category is
 Sol.  

Example: If n positive integers taken at random are multiplied together, show that the probability that the last digit of the product is 5 is  and that the probability of the last digit being 0 is 
Sol.    Let n positive integers be x₁, x₂,......, x_n. Let a = x₁, x₂.....x_n.
Let S be the sample space, since the last digit in each of the numbers, x₁, x₂......,x_n can be any one of the digits 0, 1, 2, 3, ......... , 9 (total 10)      
 ∴ n(S) = 10_n
Let E₁ and E₂ be the events when the last digit in a is 1, 3, 5, 7 or 9 and 1, 3, 7 or 9 respectively
∴ n(E₁) = 5ⁿ and n(E₂) = 4ⁿ and let E be the event that the last digit in a is 5.
n(E) = n(E₁) – n(E₂) = 5ⁿ – 4ⁿ. Hence required probability P(E) =
Second part :Let E3 and E4 be the events when the last digit in a is 1, 2, 3, 4, 6, 7, 8 or 9 and 0 respectively. Then n(E₄) = n(S) – n(E₃) – n(E) = 10ⁿ – 8ⁿ – (5ⁿ – 4ⁿ) = 10ⁿ – 8ⁿ – 5ⁿ + 4ⁿ
∴  Required probability P(E₄) =
 

Example: A bag contains n white and n red balls. Pairs of balls are drawn without replacement until the bag is empty. Show that the probability that each pair consists of one white and one red ball is 2n/(2nCn).
 Sol.    Let S be the sample space and E be the event that each of the n pairs of balls drawn consists of one white and one red ball.
   

Example: Three vertices out of six vertices of a regular hexagon are chosen randomly. The probability of getting an equilateral triangle after joining three vertices is  
                         
                           
 Sol.  The total no. of cases = ⁶C₃ = 20. As shown in the figure only two triangles ACE and BDF are equilateral. So number of favourable cases is 2. Hence the required probability =