## Terms

• Sample Space: The set of all possible values of a random variable is called as sample space
• Examples: Sample space for
• die roll S = {1,2,3,4,5,6}
• Coin Flip S = { H, T }
• Cardinality of Set: The size of the elements in a set is called as cardinality

## Probability

• How likely an event is to occur

• Probability of an event is a number that is

• A value between 0 and 1
• O corresponds to "unlikely" or almost never
• 1 correspons to "almost surely"
• Represent `P(event) = value`

• Probability is sometime expressed as percentage (0% and 100%)

• Rules of Probabilty

• Probability of event A is greater than or equal to zero `P(A) >= 0`
• Probability of Sample Space is one `P(S) = 1`
• If A1,A2,A3 … are the sequence of mutually exclusive events
``````P(A1 U A2 U A3 U....) = U A(i) = Σ P(A )
i
``````
• Examples: Consider a Coin Flip

• S = { H, T}
• P(H) = P(T) = 1/2
• P(H and tail) = 0
• P(H or T) = P(S) = 1
• Discrete Vs Continuous Probability

• Discrete = finite set of unique values
• Coin toss, die roll, number of cars in house hold
• P(E) = 1/N where N is number of possible Events
• Continuous = infinite set of values
• Person height, amount of rainfall in a day
• Instead, better to measure intervals
• Counting Methods

• Counting Simple Points
• Consider a coin Flip S = {H, T}
• Flip a coin 3 Three times (Sampling with replacement) How many different arrangements are possible?
• Direct approach: list all the arrangements and count them
``````HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
``````
• How many arrangements begin or end with a HEAD
• S(H??) U (S ??H) = HHH, HHT, HTH, HTT, THH, TTH = 6
• Multiplication Rule
• We need to travel from A to C through B
• Three Ways from A to B `AB = {AB(1), AB(2), AB(3) }`
• Two ways to get from B to C `BC = { BC(1), BC(2)}`
• How many ways of getting from A to C
• Clearly number of options is #AB * #BC => 3*2 = 6
• Example: A container has three balls: Red, Blue, Yellow. You draw each of the balls in turn without replacement. How many possible ways/arrangements are there

• First draw => 3 possible options
• Second draw => 2 possible options (one is already draws)
• Third draw => 1 possible option
• Total number of possible draws is => 3 * 2 * 1 => 6
• Permuation:

• Given a set of elements all distinct arrangement s of the elementes are called the permuatation of a set
• The number of ways you can arrange N elements of N possibilies
``````P(n,n) => N * N-1 * N-2 ....* 2* 1 = N!
``````
• Suppose we have 5 balls (RGBWO) and you draw 3 balls without replacement. How many possible arrangments are there
``````5 * 4 * 3 => 60

5 * 4 * 3 * 2 * 1     5!
----------------- = -------
2*1            (5-3)!
``````
• Number of possible arrangements with K elements out of n elements
``````         n!
P(n,k)= -----
n-k!
Where P represent Permuations
``````
• Example:
• Number of unique 5 letter words that can be arranged from letters {a,b,c,d,e} => 5^5 (5Pow(5)) => 3125
• Number of unique letters if each letter appears only once => 5! => 120
• Number of 3-letter words, each letter repeats only once => P(5,3) => 5!/(5-3)! => 60
• Combinatorics:

• Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures
• Given a bag of balls B = {R,G,B,W,O}
• If we pull out 2 balls the number of arrangements => 5!/3! = 20
• What if we don’t care about order i.e => {R,B} and {B,R} to be same pick, How can we calcualte the number of unique picks
• The number of picks including duplicates => P(5,2).
• Each pick has 2! possible arrangments
• C (n,k) = P(n,k)/k! = n!/(k! * (n-k)!)
• C(5,2) => 5!/ (3!2!) => (120 / 6 2) => 10
• Binomial Coeffecients

• Expressed as
• Example: You toss a coin 10 times and record the result, What is probability of getting exactly 4 heads
• Total number of => POW(2,10)
• Each arrangement is a choice to get 4 heads amount 10 tosses
``````C(10,4)
``````
• P(4 heads) = (C(10,4))/POW(2,10) => 0.205
• Example: You toss a coin 10 times and record the result, What is probability of getting exactly 4 heads or less
• The value is 0.377
• Example: A class has 15 girls and 30 boys. Pick 10 children at random. What the probability that you will pick exactly 3 girls