# Probability - Shortcuts and Questions with Answers

Probability is one of the important chapter for IBPS and other competitive exams like CAT. Questions in this chapter are relatively easier than other chapters. Most of the time set of 4-5 questions is asked in competitive exams which can be solved in minutes. I have tried my best to provide shortcut tricks for my readers. In case you find any problem then please comment below. Before reading this article read Permutation and Combination concept.

### My previous posts on Quantitative aptitude :-

Time and Work
Data Sufficiency

Probability = Occurrence of event/Total possible events

## Simple probability problem

Simplest questions asked in bank exams are asked about deck of cards, dice and coins. There is no co-occurrence of events.

Lets take an example :-

Question -One card is drawn from a deck of 52 cards. Find the probability that it is a black queen or king.

Simple solution to this problem is 2/52 = 1/26

## Venn Diagram

Question - One card is drawn from a deck of 52 cards. Each card is equally likely to be drawn. Find the probability that card drawn is either black or queen.

In this question we are going to use a simple formula

P(AB) = A ∪ B ∩ AB

Don't be confused with the formula. Give a look to this diagram

In a deck of cards there are 26 black cards and 4 queens. But 2 black cards are queen. These cards are part of grey and brown circles in above diagram. As pink region is added twice, we need to deduct it once.

So the solution is  26 + 4 - 2 = 28

Probability (AB) = 28/52 = 7/13

## Independent Events

In case you need to find probability of happening of two independent events we use multiplication theorem.

P(AB) = P(A) * P(B)

Lets understand this theorem with help of an example

Question - An unbiased coin is tossed twice. Find the probability of getting 1,2,3 or in the first toss and 4,5 or 6 in second attempt ?

Solution - Probability of getting 1,2,3 or 4 in first toss = 4/6 = 2/3
Probability of getting 4,5 or 6 in second toss = 3/6 = 1/2

Now both events are independent and we will use multiplication theorem

1/2 * 2/3 = 1/3

## Important Probability questions set - Venn diagram

This question is frequently asked in the exams. Even in IBPS 2012 and SBI 2013, this question appeared with slight difference in amounts. Still most of the candidates are unable to solve this.

Question - There are 200 students in commerce batch of Khalsa college. Out of them 100 play cricket, 50 play hockey and 60 play basketball. 3o students play both cricket and hockey, 45 students play both cricket and basketball and 35 students play both hockey and basketball.

1) What is the maximum number of students who play all the games ?
2) What is minimum number of students who play all three games ?
3) What is the maximum number of students who play at least one game ?
4) What is the minimum number of students playing at least one game ?

Solution - To make the question simple, draw a Venn diagram and let x = number of students who play all games. Complete the diagram.

Solution 1
What is the maximum value of x ?
Consider equation 30 - x, it states that maximum value of x = 30
In case x>30, for example 32 then 30 - x will be negative and number of students can't be negative. So answer is 30.

Solution 2
What is the minimum value of x ?
consider x-20 is possible as negative value of students is not possible. It is possible to deduct 20 from x.
So minimum value of x = 20

Solution 3 and 4
For 3rd and 4th questions we will add everything :-

25 + x + x - 15 + x - 20 + 45 - x  + 35 - x + 30 - x + x =
= 100 + x
Here we add maximum and minimum value of x
so answer is 120 and 130 respectively. 