Conditional Probability

• The probability that event B will occur given that A will occur (or has occurred) is denoted P(B|A) (read the probability of B given A)

• For example, if I have three red slips of paper and 2 blue and I want to draw two slips. P(R on 2nd |B on 1st ) = 3/4

• Given the table above what is the probability that a 26 year old will survive to age 46?

• P(alive at 26 and 46)/P(alive at 26) = 0.1/0.25 =40%

Age 0 6 16 26 36 46 56 66 76

Survivors 100% 64% 40% 25% 16% 10% 6% 3% 1%

• Thus events A and B are independent if and only if P(A and B) = P(A) *P(B)

The Basic Counting Principle

• If a certain experiment can be performed in r ways, and corresponding to each of these ways another experiment can be performed in k ways, then the combined experiment can be performed in r * k ways.

Permutations• Suppose you are selecting k objects from n

objects and the order of selection matters, there are n * (n-1) * ... *(n - k +2) * (n - k + 1) distinct sequences that can be obtained.

• Think of filling k boxes sequentially. You have n choices for the first, n-1 for the second, n-2 for the third and so on.

Permutations• Examples• # ways 5 cards can be dealt (where order matters)

from deck of 52 cards is 52*51*50*49*48• A president, secretary and treasurer are to be

selected from a club that has 20 members. How many different committees can be formed.

20*19*18

The Rule of Combinations

• Suppose that we did not care who was assigned what position on the committee.

• In the list of 20*19*18 possible choices we had, each set of three names appears six times. (Think of filling 3 boxes with 3 names).

• So if position chosen does not matter there are 20*19*18/3*2 committees.

The Rule of Combinations

• When we select k objects from n objects and order does not matter this is called a combination

• For example a committee of 4 can be chosen from 30 in 30*29*28*27/4*3*2 ways.

nk

n n n n k

k k

( )( ) ( )

( ) ( )( )( )

1 2 1

1 3 2 1

Example

• If we want to know the number of 5 cards hands that can be dealt. It is

2*3*4*5

48*49*50*51*52