# Randomized Algorithms | Set 0 (Mathematical Background)

**Conditional Probability** Conditional probability P(A | B) indicates the probability of even ‘A’ happening given that the even B happened.

We can easily understand above formula using below diagram. Since B has already happened, the sample space reduces to B. So the probability of A happening becomes P(A ∩ B) divided by P(B)

Below is Bayes’s formula for conditional probability.

The formula provides relationship between P(A|B) and P(B|A). It is mainly derived form conditional probability formula discussed in the previous post.

Consider the below formulas for conditional probabilities P(A|B) and P(B|A)

Since P(B ∩ A) = P(A ∩ B), we can replace P(A ∩ B) in first formula with P(B|A)P(A)

After replacing, we get the given formula. Refer this for examples of Bayes’s formula.

**Random Variables: **

A random variable is actually a function that maps outcome of a random event (like coin toss) to a real value.

Example :

Coin tossing game : A player pays 50 bucks if result of coin toss is "Head" The person gets 50 bucks if the result is Tail. A random variable profit for person can be defined as below : Profit = +50 if Head -50 if Tail Generally gambling games are not fair for players, the organizer takes a share of profit for all arrangements. So expected profit is negative for a player in gambling and positive for the organizer. That is how organizers make money.

**Expected Value of Random Variable : **

Expected value of a random variable R can be defined as following

E[R] = r1*p1 + r2*p2 + ... rk*pk ri ==> Value of R with probability pi

Expected value is basically sum of product of following two terms (for all possible events)

a) Probability of an event.

b) Value of R at that even

Example 1:In above example of coin toss, Expected value of profit = 50 * (1/2) + (-50) * (1/2) = 0Example 2:Expected value of six faced dice throw is = 1*(1/6) + 2*(1/6) + .... + 6*(1/6) = 3.5

**Linearity of Expectation:**

Let R_{1} and R_{2} be two discrete random variables on some probability space, then

E[R_{1}+ R_{2}] = E[R_{1}] + E[R_{2}]

For example, expected value of sum for 3 dice throws is = 3 * 7/2 = 7

Refer this for more detailed explanation and examples.

**Expected Number of Trials until Success**

If probability of success is p in every trial, then expected number of trials until success is 1/p. For example, consider 6 faced fair dice is thrown until a ‘5’ is seen as result of dice throw. The expected number of throws before seeing a 5 is 6. Note that 1/6 is probability of getting a 5 in every trial. So number of trials is 1/(1/6) = 6.

As another example, consider a QuickSort version that keeps on looking for pivots until one of the middle n/2 elements is picked. The expected time number of trials for finding middle pivot would be 2 as probability of picking one of the middle n/2 elements is 1/2. This example is discussed in more detail in Set 1.

Refer this for more detailed explanation and examples.

**More on Randomized Algorithms:**

- Randomized Algorithms | Set 1 (Introduction and Analysis)
- Randomized Algorithms | Set 2 (Classification and Applications)
- Randomized Algorithms | Set 3 (1/2 Approximate Median)

All Randomized Algorithm Topics

This article is contributed by **Shivam Gupta**. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above