Probability-2

Recap
Why should we learn Probability?
Formulating questions in terms of probability
Building the probability model
Four-step Method
Uniform sample spaces
Counting

Today’s Objectives
Counting subsets of a set
Conditional Probability
Independence
Total Probability Theorem
Baye’s theorem
Random variables

Why Count Subsets of Set?
Example:
Suppose we select 5 cards at random from a deck of 52 cards.
What is the probability that we will end up having a full house?
Doing this using the possibility tree will take some effort.

Conditional Probability
An Interesting Kind of Probability Question
“After this lecture, when I go to UI canteen for lunch, what is the probability that today they will be serving biryani (my favorite food)?

Biryani ☺

Conditional Probability
Of course, the vast majority of the food that the cafeteria prepares is NEITHER delicious NOR is it ever biryani (low probability).
But they do cook dishes that contain rice, so now the question is “what’s the probability that food from UI is delicious given that it contains rice?”
This is called “Conditional Probability”

Conditional Probability
What is the probability that it will rain this afternoon, given that it is cloudy this morning?
What is the probability that two rolled dice sum to 10, given that both are odd?
Written as
P(A|B) – denotes the probability of event A, given that event B happens.

Conditional Probability
So, how to answer the “Food Court” question?

Why Do Tree Diagrams Work?
We have solved multiple probability problems using tree diagrams
Let’s think for a moment about “why do tree diagrams work?”
The answer involves conditional probabilities
In fact, the probabilities that we have been recording on the edges of a tree diagram are conditional probabilities
More generally, on each edge of a tree diagram, we record that the probability that the experiment proceeds along that part, given that it reaches the parent vertex

Why Do Tree Diagrams Work?
Let’s look the upper most edges of the probability tree for the previous example!

Why Do Tree Diagrams Work?

Why Do Tree Diagrams Work?

Why Do Tree Diagrams Work?
“So the Product Rule is the formal justification for multiplying edge probabilities in a probability tree to get outcome probabilities”

What Independence Really Means?
Are these events independent?
A
B

What Independence Really Means?
Thus being dependent is completely different from being disjoint!

What Independence Really Means?
Thus being dependent is completely different from being disjoint!
Two events are independent, if the occurrence of one does not change our belief about the occurrence of the other.

What Independence Really Means?
Thus being dependent is completely different from being disjoint!
Two events are independent, if the occurrence of one does not change our belief about the occurrence of the other.
Typically the case when the two events are determined by two physically distinct and non-interacting processes.
Getting heads in a coin toss and snowing outside

Independence---Cont.
Generally, independence is an assumption that we assume when modeling a phenomenon.
The reason we so-often assume statistical independence is not because of its real-world accuracy
It is because of its armchair appeal: It makes the math easy
How does it do that?
By splitting a compound probability into a product of individual probabilities.
(Note for TAs: Include example of Independence assumption in tutorials)

Total Probability Theorem
Take a look at the figure below
A1
A2
A3
B

Total Probability Theorem
Take a look at the figure below
A1
A2
A3
B

Total Probability Theorem
Take a look at the figure below
A1
A2
A3
B

Total Probability Theorem
Take a look at the figure below
A1
A2
A3
B

Total Probability Theorem
Take a look at the figure below
A1
A2
A3
B

Total Probability Theorem
Where do we use it?
Baye’s Theorem!

Medical Testing Problem
Let’s assume a “not-so-perfect” test for a medical condition called BO suffered by 10% of the population
The test is not-so-perfect because
90% of the tests come positive if you have BO
70% of the tests come negative if you don’t have BO
If we randomly test a person for BO, and if the test comes positive, what is the probability that the person has BO.

Probability Tree
A: The test came positive
B: The person has BO
BO is suffered by 10% of the population
If someone has BO, there is a 90% chance that the test will be positive
If someone does not have the condition, there is a 70% chance that the test will be negative.

Conditional Probability Tree---Cont.
Surprising, Right!
So if the test comes out positive, the person has only 25% chance of having the diseases
Conclusion:
Tests are flawed
Tests give test probabilities not the real probabilities

Bayes Theorem---Cont.
A Posteriori Probabilities
For example:
The probability that it was cloudy this morning, given that it rained in the afternoon.
Mathematically speaking, there is no difference between a posteriori probability and a conditional probability.

Random Variables
So far, we focused on probabilities of events.
For example,
The probability that someone wins the Monty Hall Game
The probability that someone has a rare medical condition given that he/she tests positive

Random Variables
But most often, we are interested in knowing more than this.
For example,
How many players must play Monty Hall Game before one of them finally wins?
How long will a weather certain condition last?
How long will I loose gambling with a strange coin all night?
To be able to answer such questions, we have to learn about “Random Variables”

Random Variables---Cont.
“Random Variables” are nothing but “functions”
A random variable R on a probability space is a function whose domain is the sample space and whose range is a set of Real numbers.

Random Variables---Cont.
“Random Variables” are nothing but “functions”
A random variable R on a probability space is a function whose domain is the sample space and whose range is a set of Real numbers.
Let’s look at this example!
Tossing three independent coins and noting
C: the number of heads that appear
M: 1 if all are heads or tails, 0 otherwise
If we look closely, we will see that C and M are in fact functions that map every outcome of the experiment to a number.

Expected Value
Weighted average of the values of a random variable
Provides a central point for the distribution of the values of a random variable
We can solve many problems using the notion of expected values
How many heads are expected to appear if a coin is tossed 100 times?
What is the expected number of comparisons used to find an element in a list using the linear search?

Variance
Consider the following two gambling games:
Game A: You win $2 with probability 2/3 and lose $1 with probability 1/3.
Game B: You win $1002 with probability 2=3 and lose $2001 with probabil- ity 1=3.
Which game would you play?

Variance
Let’s compute the expected return for both games:

Variance
Game A: You win $2 with probability 2/3 and lose $1 with probability 1/3.

Variance
For game B
Intuitively, this means that the payoff in Game A is usually close to the expected value of $1, but the payoff in Game B can deviate very far from this expected value – high variance means high risk.

Standard Deviation
Because of its definition in terms of the square of a random variable, the variance of a random variable may be very far from a typical deviation from the mean.

Standard Deviation
For example, in Game B above, the deviation from the mean is 1001 in one outcome and -2002 in the other. But the variance is a whopping 2,004,002
The problem is with the “units” of variance.
If a random variable is in dollars, then the expected value is also in dollars, but the variance is in square dollars