STAT 100 Lecture 7: Random Samples and Combinations, Study notes of Probability and Statistics

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STAT 100 Lecture 7:

Random Samples

Nate Strawn

http://www.math.umd.edu/∼nstrawn/

Last Time...

(^1) Definition of Conditional Probability

P(A|B) =

P(AB)

P(B)

(^2) Definition of Independence

P(A|B) = P(A)

or P(AB) = P(A)P(B)

How Do We Count Combinations?

How many ways can we choose two people from a group of five?

How Do We Count Combinations?

This can become a very tedious exercsise!

Another Way to Count

number of pairs in a set of 5 =

The Rule of Combinations

Definition The number of possible choices of r objects from a group of N distinct objects is

(N

r

and is read “N choose r.” We have that ( N r

N!

r !(N − r )!

Formula ( N r

N × (N − 1) × · · · × (N − r + 1) r × (r − 1) × · · · × 2 × 1

A Useful Identity

When we choose r objects from a set of N objects, we have implicitly chosen a collection of N − r objects as well. These are the objects that we did not “choose.” So, every choice of r objects corresponds to a choice of N − r objects.

Proposition ( N r

N

N − r

Combinations Example

4.83 Evaluate: (a)

( 6 3

)

(b)

( 10 3

)

(c)

( 22 2

)

(d)

( 22 20

)

(e)

( 30 3

)

(f)

( 30 27

)

Another Example

4.85 Of 10 available candidate for membership in a university committee, 6 are men and 4 are women. The committee is to consist of 4 persons. (a) How many different selections of the committee are possible? (b) How many selections are possible if the committee must have 2 men and 2 women? (c) If the selection of the committe is random, what is the probability that the committee consists of exactly 2 men and 2 women?

For Next Time

MINITAB Project 01 Due Today at 4pm!

Read Section 5.1, 5.2, and 5.3 from Johnson and Bhattacharyya

Group Problems:

Group 1 2 3 4 5 Problem 4.84 4.101 5.1 5.3 5. Group 6 7 8 9 10 Problem 4.84 4.101 5.1 5.3 5.