1. P(A) = --------------------
Probability
Review of Basic Facts
Certainty for an event (a set of outcomes) to occur is measured by a number between 0 and 1. This number, called the probability of that event, is the ratio of the number of outcomes in this set to the number of all possible outcomes.
n(A)
P(A) =
n(U)
where A denotes the set of outcomes that comprises the event and U denotes the set of all the outcomes. Note that this ratio is defined only when both sets are finite.

2. If the set U of all possible outcomes is an infinite set, one must first define a suitable measure to the set U (outside the scope of this course…..)
Empirical probability is obtained by first drawing a sample U ( usually far smaller than the entire set of all the possible outcomes) and by counting the outcomes in the event only within the sample set. If the theoretical probability is known, the empirical probabilities obtained in this way approach the theoretical probability as the sample size approaches the population size. (Law of Large Number)

3. Basic Properties of Probabilities
P( { } ) = and P( U) = 1
P(A U B ) = P(A) + P(B)
if A and B are disjoint.
P(~A)= 1- P(A)
[ The chance that A does not occur is 1 minus the chance that A occurs.]

4. Conditional Probability
Let A and B be two events. Suppose we know that a certain outcome is in the set B. In this case the probability that this outcome is in A is not n(A)/n(U).
Example: Suppose you already know that the result of a dice throwing is an even numbered face. What would be the probability that this result was 3? It is not 1/6.
Because of the knowledge that the outcome is in B, the population size must be adjusted to B and the set A should be adjusted to A^B. The probability computed with such adjustments is called the conditional probability and is denoted by P(A|B)
n(A^B)
P(A|B) =
n(B)
A useful result
P(A^B)= P(A|B) P(B).

5. Examples (conditional probability)
A jar contains 3 blue balls, 1 blue cube, and 2 red balls and 1 blue cube.
After drawing one object randomly, one finds out it is a cube. In this situation, what is the probability that this object is red?
After drawing one object randomly, one finds out that it is blue. What is the probability that it is a ball?

6. Independent Events
Two events (sets) A and B are said to be independent if the knowledge that an outcome is in one of these two events does not influence the probability that this outcome is in the other set.
[Example] A jar has 3 red balls and 2 blue balls. Let U be the set of outcomes from drawing a ball from the jar, then put it back
and draw a ball again. Let A be the event consisting of all the outcomes in which the first ball was blue. Let B be the event consisting of all the outcomes in which the second ball was red.
Then, obviously, knowing that the first drawing was red or blue does not influence the probability of drawing red or blue in the second drawing. So, A and B are independent.
Formal Definition of Independence: We say A and B are independent if
P(A) = P(A|B)

7. Here A and B are independent
In this example, A and B are not independent

8. Here is a good example making a clear contrast between the dependent events and independent events. Suppose a jar has 3 red and 2 blue balls.
Q1. Suppose you draw a ball randomly, find that it is red, replace it, and draw a ball again. What is the probability that the second draw is a blue ball?
Q2. Suppose you draw a ball randomly, find that it is red, and draw a ball again without replacing the first ball. What is the probability that this second draw is a blue ball?

9. If A and B are independent, then
P(A)= P (A|B)= n(A^B)/ n(B)
But,
P(A^B)= n(A^B)/n(U)
P(B)= n(B)/n(U)
So, we have P(A)= P(A^B)/P(B), or
P(A^B)= P(A) P(B)

10. Independent events (an application example)
The probability that South Florida will be hit by a hurricane in any single year is 0.25.
What is the probability that South Florida is hit by a hurricane in 2 consecutive years?
What is the probability that no hurricane hits for next 10 years ?
3. What is the probability that at least one hurricane hits within next 10 years?

11. Dependent Events (example)
A jar contains 10 balls, 3 of which are red.
Suppose you pick 2 balls one by one without replacing them. Let
A= Set of outcomes in which the first ball is red
B= Set of outcomes in which the second ball is red
Let us compute P(A^B).
P(A^B)= P(B|A) * P(A) = (2/9) * (3/10) = 1/15.

12. Disjoint Event
If two events A and B have empty intersection as sets, we say that A and B are disjoint as events. Disjoint events cannot happen together. So, if an outcome is in A, we are certain it is not in B and vice versa.
If A and B are disjoint, then
P(A U B ) = P(A) + P(B)

13. Disjoint events are always dependent
An event and itself is always dependent.
Since A and ~A are disjoint, we have
P(A)+ P(~A) = P(A U ~ A)
= P(U)
= 1.
Therefore P(~A)= 1- P(A)