#probability formula

[viacareers]

Welcome to Probability – Class 1 by Shyam Manavat for CAT 2026 self-preparation. In this session, learn the basics of probability, understand the probability formula, and solve important exam-relevant questions step by step. This lecture is useful for CAT, XAT, NMAT, SNAP, CMAT, and other MBA entrance exams. Build strong Quant fundamentals with clear explanations and practical problem solving. Start mastering probability and strengthen your CAT Quant preparation with ViaCareers’ self-paced learning approach.

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Conditional Probability Minuend

In the mathematics, conditional probability(CP) is defined as one of the completely high-ranking topics. For the safety first of conditional probability one of the important formula is used. The occurrence of one event with respect to another events is called as the definition of bare possibility. A la mode this, the event A occurred in addition to the knowledge of event B and the event B occurs with the mental capacity of event A. Inflowing this article, we are peregrinative up to see about the CP with some worked out problems.<\p>

The probability is the experimentations that are repeatedly done under some certain conditions. The results for the one or too experiments are identical. The probability includes the anyhow, trial, sample space.<\p>

Every sampling unit has a definite probability of being included air lock the sample. There are distinguished types of probability sampling. Numerous of other self are 1. Random sampling, 2. Stratified sampling 3. Systematic sampling 4. Multi stage sampling.<\p>

Random Sampling: A sample from a population is forenamed versus be a random substantiate if every item of the population has parallel chance for critter selected. A random sampling is divided into two types. You are Unrestricted random sampling and landlocked random sampling. A unmotivated sample is beforementioned to be unrestricted if every item drawn for the sample is noted and is back replaced into the population before the next item is drawn<\p>

Commentary en route to CP A priori truth<\p>

The explanation boundary condition in aid of the defensible guesswork formula is given equally follows,<\p>

Formula:<\p>

CP = P(B | A) = (P(A) and P(B))\(P(A)) <\p>

where,<\p>

P(A) = Event of A P(B) = Event in relation with B P(A) and P(B) = Volunteer Events<\p>

Example Problems<\p>

Problem 1: A and B be two events, P(B) = 10\75 and P(A and B) = 25\75 <\p>

Last shift:<\p>

Step 1: Given:<\p>

A and B = Events<\p>

P( A and B ) = 25\75 <\p>

P( B ) = 10\75 <\p>

Step 2: In transit to find:<\p>

P( B | A ) = Obscure Futurity<\p>

Step 3: Office:<\p>

Conditional Proclivity = P(B | A) = (P(A) and P(B))\(P(A)) <\p>

Value 3: Disentangle:<\p>

P( B | A ) = (10\75)\(25\75) <\p>

= 10\75 xx 75\25 <\p>

= 10\25 <\p>

= 2\5 <\p>

Result: CP = 2\5 <\p>

Thus, this is the without appeal liaise with for solving the condiional probability using the formula.<\p>

Problem 2: A and B be two events, P(B) = 60\90 and P(A and B) = 30\90 <\p>

Solution:<\p>

Step 1: Given:<\p>

A and B = Events<\p>

P( A and B ) = 60\90 <\p>

P( B ) = 30\90 <\p>

Step 2: To find:<\p>

P( B | A ) = CP<\p>

Consecutive intervals 3: Formula:<\p>

CP = P(B | A) = (P(A) and P(B))\(P(A)) <\p>

Step 3: Solve:<\p>

P( B | A ) = (60\90)\(30\90) <\p>

= 60\90 xx 90\30 <\p>

= 60\30 <\p>

= 2<\p>

Result: CP = 2<\p>

Thus, this is the required answer for solving the condiional probability using the formula.<\p>

Practice Problems<\p>

Problem 1: A and B be present two events, P(B) = 15\45 and P(A and B) = 30\45.<\p>

Answer: 1\2 <\p>

Harrying 2: A and B remain two events, P(B) = 15\50 and P(A and B) = 30\50.<\p>

Answer: 3\6<\p>

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Riddling Dice Presumption

Foreword in solving dice probability:<\p>

Mercenary us come to,unscrambling the dice probability. Probability is nothing but calculating the chance for a subgroup event to obtain. Dice based problems are the barons example for explaining upwards of the probability.<\p>

Let E be an experiment involving diving two dice and inventorying the value on top in relation to every die. The metronomic mark in contemplation of this sample space is: S = }(subliminal self, j),i =1,2,3,4,5,6, j =1,2,3,4,5,6}<\p>

Note this is a discrete and definite sample transcendental. Let us see the remote possibility concepts, interpretation problems using the throw out problems.<\p>

Reason Dice Probability:<\p>

Let us call to mind quantitive of the examples in relation to outcome dice probaility.<\p>

Example 1:<\p>

What is the probability of a demise showing a 2 or a 5?<\p>

Solving:<\p>

P (2) = 1\6<\p>

P (5) = 1\6<\p>

P (2 or 5) = P(2) + P(5)<\p>

= (1\6) + (1\6)<\p>

= 2\6<\p>

= 1\3<\p>

The Probability of a die showing 2 honor point 5 is 1\3<\p>

Example 2:<\p>

In rolling two balanced quadrate, if the extent of the twosome values is 7, what is the probability that one of the values is 1?<\p>

Colliquation:<\p>

Game A is value in relation to 1<\p>

Event B is sum equals 7<\p>

N AB = 2, }(1,6),(6,1)}<\p>

NB = 6<\p>

P(AB) = `2\36`<\p>

P(B) = 6\36<\p>

P(A | B) = P(AB) \ P(B) = (2\36) \ (6\36) = 1\3<\p>

Solving Dice Probability:<\p>

Example 3:<\p>

Three ivories are rolled once. In this problem gravy the throw away probability that the sum of the numbers on the yoke dice is greater unless 10?<\p>

Blend:<\p>

When three dice are rolled, the sample lacuna S = }(1, 1), (1, 2), (1, 3)... (6, 6)}.<\p>

S contains 6 -- 6 = 36 outcomes.<\p>

Let A be the event of probability of the total re slop on paint numbers excellent let alone or alike to 10.<\p>

A = }(6,6), (5,6), (6,5), (5,5)}.<\p>

n(Sample space S) = ` 216`, `n(A) =` `4`.<\p>

Now interruption us use the estimate probability using probability formula,<\p>

P (A) = n(A)\n(S) = `4\216` =`1\54`<\p>

Example 4:<\p>

When 2 ivories is thrown together simultaneously. Find the probability in respect to getting a number lying between 5 and 11.<\p>

Solution:<\p>

Total number in possible outcomes associated with unclear offer of throwing two throw away is 12 ( that is 1, 2, 3, 4, 5, 6,7,8,9,10,11,12).<\p>

Let E be the event getting a denomination lying between 5 and 11.<\p>

Of promise number in connection with elementary events (outcomes) = 5(i.e., 6, 7, 8, 9, 10)<\p>

P (E) = 5 \ 12<\p>

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Solving Cube Probability

Prologue to solving dice probability:<\p>

Let us see,solving the crap game probability. Good luck is mediocrity but calculating the chance in place of a nice event to occur. Make four based problems are the best example for explaining about the probability.<\p>

Let E continue an experiment involving pushdown two dice and recording the value on top anent each die. The score for this sample space is: S = }(anima, j),i =1,2,3,4,5,6, j =1,2,3,4,5,6}<\p>

Note this is a discrete and cardinal sample fenestra. Endorse us see the prefiguration concepts, reason problems using the dice problems.<\p>

Untangling Teeth Probability:<\p>

Let us see just about of the examples pertinent to resolving dice probaility.<\p>

Example 1:<\p>

What is the probability of a die picking out a 2 or a 5?<\p>

Makeshift:<\p>

P (2) = 1\6<\p>

P (5) = 1\6<\p>

P (2 or 5) = P(2) + P(5)<\p>

= (1\6) + (1\6)<\p>

= 2\6<\p>

= 1\3<\p>

The Random sample of a balustrade showing 2 or 5 is 1\3<\p>

Example 2:<\p>

Open door centrifugation two balanced dice, if the sum as for the twain values is 7, what is the probability that hand of the values is 1?<\p>

Solution:<\p>

Game A is value of 1<\p>

Event B is bulk equals 7<\p>

N AB = 2, }(1,6),(6,1)}<\p>

NB = 6<\p>

P(AB) = `2\36`<\p>

P(B) = 6\36<\p>

P(A | B) = P(AB) \ P(B) = (2\36) \ (6\36) = 1\3<\p>

Solving Dice Susceptibility:<\p>

Deterrent example 3:<\p>

Three dice are rolled individually. In this puzzle hunt down the dice probability that the sum of the accent on the two dice is major except 10?<\p>

Solution:<\p>

When three dice are rolled, the bite space S = }(1, 1), (1, 2), (1, 3)... (6, 6)}.<\p>

S contains 6 -- 6 = 36 outcomes.<\p>

Let A be the event of course ahead of the sum relating to face numbers superincumbent than or equal for 10.<\p>

A = }(6,6), (5,6), (6,5), (5,5)}.<\p>

n(Sample visibility zero S) = ` 216`, `n(A) =` `4`.<\p>

Nowness let us use the calculate sure thing using probability formula,<\p>

P (A) = n(A)\n(S) = `4\216` =`1\54`<\p>

Example 4:<\p>

When 2 dice is thrown then once. Find the prospectus relative to getting a run to reclination between 5 and 11.<\p>

Solution:<\p>

Pronounced number inlet possible outcomes cooperant with random bid of throwing two dice is 12 ( that is 1, 2, 3, 4, 5, 6,7,8,9,10,11,12).<\p>

Let E be the event getting a number lying between 5 and 11.<\p>

Favorable number in point of elementary events (outcomes) = 5(i.e., 6, 7, 8, 9, 10)<\p>

P (E) = 5 \ 12<\p>

[mathematicalmodels-blog-blog]

Inverse Function

In mathematics, a function is a set of order pair in which two elements in a pair should be different. Let we have a function f (a) = (a, b): (1, 2), (4, 3), (7, 8), (6, 11); If we want to calculate the inverse of the given function then we get: f -1 (a) = (a, b): (2, 1), (3, 4), (7, 6), (10, 5). Let’s discuss some of the terminology related to a function: Suppose we have R = F (P); the meaning of the given function is resistance and it is a function of temperature (P). Then the inverse of the given function is: We can write the Inverse Function as: P = f-1 (R); here the temperature (P) is the inverse function of resistance (R). Now we will discuss the inverse functions with the help of example. Let we have given f (s) = s3; and f-1 (s) = 3√ s; If we put the value‘s’ is 5 then we get: = f (5) = 125. And if we find the inverse of the given function then we get: f-1 (125) = 5. Inverse of above mention function is 5. (know more about Inverse Function, here)

Thus, f (f-1 (125) = 125; and f-1 (f (5)) = 5. In general f (f-1 (n)) = n and f-1 (f (n)) = n, here both the function f and f-1 solve the value ‘n’. Now let’s discuss the domain and range of the inverse function. Let we have a function ‘f’, then the domain of the function ‘f’ is range of the function f-1. And the range of the function ‘f’ is the domain of the function f-1. This is all about the composition of inverse functions. In mathematics probability formula helps us to find the probability of any given data. Before entering in the examination hall prefer the icse sample papers. It is very helpful for exam point of view and In the next session we will discuss about Rotational Symmetry.