What is the probability of knowing everything about Probability?

Today we will learn a very interesting topic and I will try my best to explain the topic so that the fear you have in your mind about this chapter will completely cut off.
So let’s start.

What is Probability?

From its name, I think it is clear that it indicates some probable values.

It is a part of mathematics, where we see some events related to our daily life and we try to find out the possibilities of occurring that event.  

Basically, after reading this chapter you will be Dr Strange.🤭(just joking)

But yeah of course here in this chapter we will find the possibilities of an event.

Many of the students think that this is a very difficult topic(like me),  but believe me, it’s not so difficult. 

But you may ask that “Why we should read probability?” More generally,

Applications of Probability:

Probability is used to find the final result in many Grambling games. Also, this is used to predict the weather of a day. 

Actually, ‘Probability’ was invented in a gambling game😂

Invention:

In a gambler’s debate in 1654 Probability was invented by two famous French mathematicians, Blaise Pascal and Pierre de Fermat. 

Introduction of some basic fundamentals:

  • Any experiment, of which all possible outcomes are known beforehand, but the exact outcome at a particular execution of the experiment is not known beforehand is called a Random experiment. For example – tossing a coin, throwing a die, etc.
  • The set of all possible outcomes of a random experiment is called its Sample space. It is usually denoted by ‘S’. And subsets of a sample space is called Event.
  • The set of one outcome is called a Simple event and a set of more than one outcome is called a compound event.
  • The probability of an event is generally defined as –

(total number of favourable outcomes to that event)/

(total number of possible outcomes of that experiment)

⇒Symbolically – P(A)=n(A)/n(S)

Basic rules of Probability:

Now let’s see some basic rules we have to remember during solving a problem.

 

  1. 0≤P(A)≤1
  2. P(A)+P(B)=1
  3. P(AUB)=P(A)+P(B)-P(A∩B)
  4. P(A)=0 means it is an impossible event
  5. P(A)=1 means it is a sure event
  6. P(Ac) =1-P(A)
  7. Let, A and B are two events related to event E. Then, A⋃B[or A+B] denotes that at least one of them will occur and P(A⋃B) denotes the probability of occurring at least one of them. A∩B[or AB] denotes that both A and B will occur and P(A∩B) denotes the probability of occurring both events.

What is an Event?

Anything that occurs or can occur, is considered as an event.

Types of event:

In probability, we will divide the events into some types according to their characteristics.

1. Events having the same properties are called equally likely events. 

Now Let,

        A1,A2,A3…An be subsets of the sample space ‘S’.

  1. if A1UA2UA3U…UAn =S, then the sets are called Exhaustive event.
  2. if Ai∩Aj=ϕ, i≠j, i≥1, j≤n, then the events are called Mutually exclusive events.
  3. If both the above hold true simultaneously the events are called mutually exclusive and exhaustive. In the other words A1, A2, A3…An is said to create a partition of S.

Let, A be an event, 

Ac is the complement of event A. It indicates just the opposite of A. Means suppose A indicates “the boy will solve all math problems”, then Ac will indicate “the boy will not solve any math problem”. 

such that n(A)=p &n(Ac)=q, then we can say that,

 

  1.  Odds in favour of A =p/q
  2. Odds against A =q/p

Clearly, it can be deduced that if P(A) =m/n, then odds in favor of A =m/(m-n) & odds against A =(m-n)/m

Conditional Probability:

Two events A and B are said to be dependent on each other if the occurrence or non-occurrence of A affects the occurrence or non-occurrence of B & vice versa.

=>The probability of occurrents of A given that event B has already happened is given by

P(A/B) =P(A∩B)/P(B)

If A&B are independent events, then clearly P(A/B)=P(A)

Therefore, P(A∩B) =P(A). P(B)

Baye’s Theorem:

Let, A1, A2, A3,…, An create a partition of the sample space ‘S’ of a certain experiment.

Let, ‘E’ be an event happening within the sample space ‘S’.

We know, A1⋃A2⋃…⋃An = S &Ai∩Aj =φ, i≠j, j≤n, i≥1

Now, E=E∩S=E∩(A1⋃A2⋃…⋃An)=(E∩A1)⋃(E∩A2)⋃(E∩A3)⋃…⋃(E∩An)

P(E)=P(E∩A1)+P(E∩A2)+…+P(E∩An)

⇒ P(E)= P(A1).P(E∩A1)+P(A2).P(E∩A2)+ … + P(An). P(E∩An)

⇒ P(E)= nΣ(r=1)P(Ar).P(E∩Ar) {r=1➝n}

Therefore,

P(E/A)=P(E∩A)/P(A) ⇒ P(E∩A) = P(A). P(E/A)

Hence Baye’s Theorem states if-

P(Ai/E)=P(Ai∩E)/P(E) = P(Ai).P(E/Ai)/ΣP(Ar).P(E∩Ar)

Now let’s talk about-

Binomial Distribution:

If repetition trials of a certain experiment are independent, then they are called as Binomial Trial or Bernoulli Trial.

Let,

Any event of an experiment is denoted by “success”. If the probability of success is ‘p’ then the probability of failure is denoted by ‘q’ such that, p+q=1.

Then the probability of exactly ‘r’ success is given by,

P(X=r)=ncr .pr.q(n-r)

Clearly, this is the general term in the expansion of (p+q)n.

The probabilities which are obtained by putting r=0,1,2,3,…,n comprise the probability distribution table and the sum of all those terms clearly equals (p+q)n that is 1.

If there are n then-

 

  1. Mean=np
  2. Variance = npq

Till now I hope you understood all these things.

Now let’s see some problems.

Problems:

Q1: Find the chance of throwing 1) four and 2) even number with an ordinary six-faced die.

Ans: 

Q2: Find the probability that a non-leap year should have 53 Saturdays.

Ans: 

Q3: The probability that candidates A and B fails in an examination is 0.3 and 0.4, respectively. Find the probability that 1) both A and B fail and 2) either A or B fails.

Ans:

Q4: A five-digit number is formed by the digits 0,1,2,3,4 without repetition. Find the probability that the number formed is divisible by 4.

Ans:

Q5: Four cards are drawn from a pack of cards. Find the probability that 1) all are diamonds, 2) there is one card of each suit and 3) there are two spades and two hearts. 

Ans:

Q6: There are 3 white, 2 red; 7 white, 3 red and 5 white, 3 red balls in three pouches. Now we pick one ball from a randomly chosen pouch. Then what is the probability that the chosen ball was white?

Ans:

Q7: It is known that if a man talks 4 times, he tells the truth 3 times. While playing dice with his friend he throws the dice and said that 6 shown in the dice. Find out the probability that he said the truth.

Ans:

So these are two types of examples.

Hope you understand all these things.

If you have any queries then comment below, I’ll try to clear your query. And if you like this then don’t forget to subscribe to us.

Snehasish Konger
Snehasish Konger

Hey, I'm Snehasish Konger, the guy behind this website.
Read my blogs from here-

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