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Maths Presentation
Probability
Probability
What is Probability ?
Probability is a branch of Mathematics that
deals with calculating the likelihood of a given
event's occurrence, which is expressed as a
number between 1 and 0.
Example
Playing Cards
If you draw a card from a standard deck of
cards , what is the probability of not getting a
heart?
Example
Bag with Marbles
If a bag contains 5 white and 7 red marbles,
what is probability of getting a red marble ?
Real Life Problems
Real Life Problems
- Size of independent set in a graph
- IQ Problem
- Coupon Collector Problem
Compute the lower bound on the size of the largest Independent Set of Graph.
Problem - 1
The largest independent set is
What is an independent set ?
Independent Set
Degree of vertex of a Graph
Degree of Vertex of a Graph
Cycle Graph
Largest Independent set in a Cycle Graph is of [n/2] size
where n is the total number of vertices.
Procedure 1
n = total number of vertices
I = Independent Set =
S = {v1 , v2 , v3 , …........vn }
Repeat the following till S is not empty :
- Remove the lowest index vertex v from S and add it to I.
- Remove all the neighbours of v from S if they are not already removed.
- Repeat
Procedure 2
Let's bound the expected size of I
v= a vertex or node of graph
Iv = an indicator variable which indicates whether or not v belongs to I i.e. Iv = 1 if it does belong or 0
V belongs to I if none of its neighbour appear before it in the permunation. This happens with probability 1/dv.
There must be atleast permutation for which
the size of I is at least equal to its expectation.
Example
Initially, I = { }
Let S = {2 , 6 , 7 , 5 , 4 , 3 }
Removing lowest index vertex 2 and its neighbours :
I = { 2 }
S = { 7 , 5 , 3 }
Removing lowest index vertex 7 and its neighbours :
I = { 2 , 7 }
S = { }
= = 2.66
IQ Problem...
Problem - 2
Assume that IQ scores are normally distributed, with a mean µ of 100 and standard deviation of 15. What is the probability that a randomly selected person has an IQ score greater than 110?
Normal Distribution
Normal Distribution...
Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.
Solution
Solution...
Coupon Collector Problem
Problem - 3
Motto : To collect all the coupons and win
The Problem
Coupon Collector Problem
- In probability , the coupon collector's problem describes "collect all coupons and win" contests
- It asks the following question: If each box of a brand of cereals contains a coupon, and there are n different types of coupons, what is the probability that more than t boxes need to be bought to collect all n coupons?
- The mathematical analysis of the problem reveals that the expected number of trials needed grows as O(nlog(n)).
- For example, when n = 50 it takes about 225[b] trials on average to collect all 50 coupons
Description
Figure describing the situation :
Solution
Solution
Let T be the time to collect all n coupons, and let ti be the time to collect the i-th coupon after i − 1 coupons have been collected.
Think of T and ti as random variables.
By the linearity of expectations we have :
Here Hn is the n-th harmonic number. Using the asymptotics of the harmonic numbers, we obtain :
where , is the Euler–Mascheroni constant.
Now one can use the Markov inequality to bound the desired probability:
Thank You !
Name
Roll No.
Created by
LIT2019072
LIT2019073
LIT2019074
LIT2019075
LIT2019076
LIT2019077
Anu Kumari
Ayushi Panth
Nishan Singh
Kanu priya
Rajat Napalchyal
Harsh Tiwari