Hi Folks !! In this post i will discuss about the tricks and tips that i use to solve questions based on probability and i will also discuss about where the concept of probability is used in Statistics and ML.

Note: [ This post is on probability and probability only and not on ML Algorithms that uses probability. But I assure you, after reading this post you will feel confident in reading and understanding all the Probability based ML -Algo’s like — {Naive Bayes | Bayesian Statistics etc .. } or in Stats like — {Binomial Probability Distribution etc ]. So without wasting time, lets get started .

### Basics

Formula => P(Event) = Favourable Outcomes / Total Possible Outcomes . Lets look at some examples:

1. Problem: Throwing a Dice (1 time ) — Means [1,2,3,4,5,6] ie. total possible outcomes = 6. What is the probability of getting 5 on throwing a dice ? Ans : 1 / 6 .
2. Problem: Tossing a Coin (1 time) — Means [Heads , Tails ] ie. Total possible outcomes = 2. What is the probability of getting a Head or Tail on tossing a coin? Ans: p(H)= 1/2 , p(T) =1/2
3. Problem: Bucket with different colour balls and you are selecting a particular colour ball — Means total possible outcomes = number of balls in bucket . A bucket contains 10 green balls 7 blue balls and 3 white balls. You pick up a ball from basket (1 time) . What is the probability of getting a green ball ? Ans : 10 / (10 + 7 + 3) = 0.5

Now lets make this thing a little interesting !!!

What if i say, Throw 1 dice 10 times and find the probability of getting number ‘5’ exactly 7 times . Or Throw 5 dice together and find the probability of getting a sum 25 ?? [ Answer it in the comments section :) ] Tough huh !! Not really … To answer these questions you need to know few concepts . Lets look at them closely —

### Types of Events

#### Mutually Exclusive events

1. Two events are mutually exclusive if they do not occur at same time .
2. Example : Head or Tail cannot occur at same time on tossing a coin.
3. Question: Two dice are rolled what is the probability that the sum of numbers rolled is either 5 or 9 ? Ans : P(AUB) = P(A) + P(B)

#### Independent Events

1. Two events are independent when the result of one event does not affect the other . They both are pure independent events .
2. Example: Tossing a coin 2 times one after the other are independent events .
3. Question: two dice are thrown one after the other . what is a probability of getting an even on both dice ? Ans : P(A)*P(B) = 3/6*3/6

#### Dependent Events

1. If the result of one event affects the result of other then both events are dependent.
2. Example: A basket has 10 balls (5 Blue and 5 White) and you select the first ball and keep it aside and then you select another ball from the basket .
3. Now, lets make a question out of point 2 . what is the P of getting blue both times ? Ans: P(A) * P(B,once A occurred ) = 5/10*4/9
4. P(AUB) = P(A)*P(B, once A has occurred)

### Conditional Probability

1. This is based on the concept of dependent events , where probability that an event A takes place will depend on another event B.
2. Conditional Probability of an Event B given A [represented by=> P(B/A)]
3. Formula => P(B/A) = P(A and B intersection) / P(A)
4. Example: Suppose, I wrote 2 post about “probability” on Hackernoon and found that 25% of viewers read both of my post and 42% of viewers read only my 1st post .Now, What percent of those viewers who read my 1st post also read my 2nd post? ANS : .25/.42 = P(1st intersection 2nd) / P(1st) = 60% = P(2nd/1st) ( I hope you got this one ).

### Bayes Theorem

1. It describes the probability of an event, based on prior knowledge of conditions that might be related to the event.
2. Formula : P(A/B) = P(B/A).P(A)/P(B)

Bayes theorem and Conditional Probability question style are very much inter-related. What i mean is that you can easily convert a Bayes theorem question to a one on Conditional probability and Vice versa by just changing 1 line of the entire question . Lets Look an Example :

Q. There are 2 basket with balls- BASKET-A and BASKET-B . Each basket contains 3 blue balls and 2 green balls . You select 1 ball from a Basket .What’s the probability of selecting a Blue ball ? (Conditional Probability Question)

Ans : Here, The Probability of picking Basket-A = 1/2 . Probability of picking Basket-B = 1/2 . Probability of selecting a Blue ball from Basket-A = 3/5 . Probability of selecting a Blue ball from Basket-B= 3/5 .

P( Selecting Blue Ball ) = P(blue Ball, Basket-A is selected first ) + P(blue Ball , Basket-B is selected first )

Final Ans = 3/5*1/2 + 3/5*1/2 (I hope you got this)

Converting the above conditional Probability Question into a Bayes !!

Q. There are 2 basket with balls- BASKET-A and BASKET-B . Each basket contains 3 blue balls and 2 green balls . You select 1 ball from a Basket and found that it’s a Blue Ball. What is the probability that its from Basket-A? (Bayes Question)

P(Blue Ball ) = 3/5*1/2 + 3/5*1/2 (Calculated above in question)