Probability is an important part of maths and statistics and there is no doubt that it plays a huge role in the competitive exams also . Companies like TCS,CTS,Infosys, IBM gives a great interest in this particular topic in their aptitude tests . So no more further due let's jump into the chapter.
Okay so first let me tell you this blog is just made for the aptitude tests so we cover only the basic parts of Probability because in these types of exams question does not come from complicated portions like Bayes Theorem , Binomial Distribution of probability and all this . So let's get started......
To know probability we have to know some terminologies. Such that
- Sample Space :- When we perform an experiment, then the set of S of all possible outcomes is called Sample Space.
- Event :- Any subset of a sample space is called an event.
The main goal of probability is to calculate "The likeness of an event to occur " we can summarize this concept by this small formula .
P(E)=n(E)/n(S)
[Where P(E)=Probability of Event E , n(E)= Number of time event "E" occurs, n(s)= Number of total sample space ]
Probability of an event always have a value between 1 and 0. 1 means that the the probability of the particular case is 100% and 0 means that the case will never happen. [For example the probability that sun will rise in the east is 1 and the probability of sun will rise in the west is 0.]
So 95% of questions in the exams always come from this 4 areas of probability.....
- Coins .
- Cards .
- Dice .
- Problem regarding Students / Multicolored Ball .
1) Coins :-
So there are two sides of the coins. So for one toss the sample space will be.....
S= {Head , Tail}...........Which means
The probability of getting a Head will be = 1/2 (or half of 0.5)
The probability of getting a Tail will be = 1/2 (or half of 0.5)
Now suppose we tossed two coins so the sample spaces will be....
S = ({Head,Head},{Head,Tail},{Tail, Head},{Tail, Tail})..... generally the questions regarding the coins come from this area I mean from the problems containing the toss of 2 or 3 coins.So let us solve a quick question...
Q) Suppose two coins are tossed what is the probability of getting atleast 2heads ?
Q) Suppose two coins are tossed what is the probability of getting atmost 1head ?
Now suppose we tossed three coins so the sample spaces will be....
S=({H,H,H},{H,H,T},{H,T,H},{H,T,T},{T,H,H},{T,H,T},{T,T,H},{T,T,T}).....So let's solve some questions.....
Q) Suppose three coins are tossed what is the probability of getting at least 2 heads ?
Q) Suppose three coins are tossed what is the probability of getting at most 2 heads ?
2) Dice :-
So there are 6 sides of a die . So for throwing for one single time we could get any value from 1 to 6. So the sample space is.....
S={1,2,3,4,5,6}......For example the probability of getting any of the value is 1/6 . So let's solve some questions....
Q) An unbiased die is tossed . Find out the probability of getting a multiple of 3 .
Multiples of 3 are = {3,6}
Q) An unbiased die is tossed . Find out the probability of getting a number greater than 4.
Now if you face a problem where two dice are thrown then the sample space will be the total number of possible outcomes which means (6 X 6 = 36) . Let me explain it with some examples.
Q) Suppose two dice are thrown then what is the probability of getting a total more than 7 ?
Q) Suppose two dice are thrown then what is the probability of getting two numbers whose product is even?
Now to generalize the concept of sample space you have to remember that the sample space will be (6^n) where "n" is the number of dice. I will give you exercises to practice once I finish the chapter .
3) Cards :-
Cards are the most complicated yet the easiest portion that we are going to cover in this chapter . Before going further let me clear the concepts of cards for you. So there are 4 types of card suits .....
And 4 face cards in each suit ......
Each of the suits have 13 cards so the total number of card is (13 X 4 = 52). The sample space in cards problem is not fixed sometimes you need to use 52 as sample space or sometimes you need to use 4 [ If the problem is strictly based on one of the face cards ] or sometimes you need to use both 52 or 4 or 13 as per the requirement of the question . We will see them with examples.....
Q) From a pack of 52 a card is drawn at a random, What will be the probability that the card is red or a king ?
Q) Two cards are drawn from a pack of 52 cards . What is the probability that either both are red or both are kings ?
Q) A card is drawn from from a pack of 52 cards what is the probability that it is a face card ?
Okay I think these examples are enough to understand this portion . If you still have some doubts I will give some question and answer when the chapter is over . I am sure that will help.
4)Problem regarding Students / Multicolored Balls :-
This portion is mainly basses on the selection or more precisely speaking about Combination ( From permutation-combination) . If you don't know abut combination don't worry I will make a separate blog on Permutation and Combination you can learn it from there . But for the sake of this chapter let me give you a little concept of combination .
In combination each of the different groups or selections which can be formed by taking same or all of a number of objects.[For example :- All combination form by a,b,c taking two at a time are ab,bc,ca]
The number of all combinations of "n" things taken "r" at a time is .....
You just need to remember the formula to solve the questions from this portion.
Q) A bag contains 6 black and 8 White balls. One ball is drawn at random . What is the probability that the ball drawn is white ?
Q) Four persons are chosen at a random from a group of 3 men, 2 women and 4 children . What is the probability that exactly 2 of then are children ?
Ok so this is it from the chapter hope you understand this chapter and able to solve every questions from it . Now I will give you some questions and answers whuch will clear all your remaining doubts about probability.
Q) In a simultaneous throw of a pair of dice find the probability of getting a total of 7.
Q) Two dice are thrown together . What is the probability of getting a doublet?
Q) Two dice are thrown together . What is the probability of getting a total of 10 or 11?
Q) Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at a random . What is the probability that the ticket drawn has a number which is a multiple of 3 or 5 ?
Q) From a pack of 52 two cards are drawn at a random, What will be the probability that the cards are kings ?
Q) From a pack of 52 two cards is drawn at a random, What will be the probability that one is a spade and one is a heart ?
Q) Suppose two dice are thrown then what is the probability of getting the total sum is a prime number?
Q) A box contain 5 red , 6 blue , 4 White balls . Three balls are drawn at a random. What is the probability that all balls are red ?
Q) A box contain 10 blacks, 10 white balls . Two balls are drawn at a random. What is the probability that all balls are same colored ?
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