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:

**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 .**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**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**

- Two events are mutually exclusive if they do not occur at same time .
- Example : Head or Tail cannot occur at same time on tossing a coin.
- 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

- Two events are independent when the result of one event does not affect the other . They both are pure independent events .
- Example: Tossing a coin 2 times one after the other are independent events .
- 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

- If the result of one event affects the result of other then both events are dependent.
- 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 .
- 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
- P(AUB) = P(A)*P(B, once A has occurred)

### Conditional Probability

- This is based on the concept of dependent events , where probability that an event A takes place will depend on another event B.
- Conditional Probability of an Event B given A [represented by=> P(B/A)]
- Formula => P(B/A) = P(A and B intersection) / P(A)
- 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

- It describes the probability of an event, based on prior knowledge of conditions that might be related to the event.
- 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 ball**s . 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 ball**s . 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(Basket-A | Blue Ball )= P(Blue Ball | Basket-A)* P(Basket-A)/P(Blue Ball)

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

P(Basket-A) = 1/2

P(Blue Ball | given Basket-A is selected ) = 1/2*3/5

Final Answer ( Simply put values in formula )= (3/10 * 1/2 )/(6/10)

You can also visualise the entire scenario by making a probability tree. (The image at the top) and then calculate evertyhing .

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