How likely is it that…..?

The Law of Large Numbers says that the more times you repeat an experiment the closer the relative frequency of an event will come to the theoretical probability of the event.

Let’s try an experiment: Flip a coin 20 times and record how many heads and how many tails….

Any situation involving uncertainresults is called an experiment

The set of all outcomesof an experiment iscalled the sample space.

Experiment 1: A single Coin is tolled once and theoutcome -- a head or a tail is recordedS = {h,t} and n(S) = 2

Experiment 2: Two coins a penny and a nickel aretossed simultaneously and the outcome of each coin isrecorded using ordered pair notation (penny, nickel)

S = {(H,H),(H,T), (T,H),(T,T)} and n(s) = 4

T HT

NICKEL

HH

T

Experiment 3: A single die is rolled

S={1, 2, 3, 4, 5, 6}

Experiment 4: Two dice are rolled and the sum of theirdots is recorded

Try it!

There are 11 possible outcomes. Are they

equally likely?

Experiment 4: Two dice are rolled and the sum of theirdots is recorded

S1 = {2,3,4,5,6,7,8,9,10,11,12}

S2 = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6), (2,1),(2,2),(2,3),(2,4),(2,5),(2,6), (3,1),(3,2),(3,3),(3,4),(3,5),(3,6), (4,1),(4,2),(4,3),(4,4),(4,5),(4,6), (5,1),(5,2),(5,3),(5,4),(5,5),(5,6), (6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

Experiment 4: A box containing three poker chips (onered, one blue, and one white) and two are drawn withreplacement. The chips are scrambled in between draws.

R

B

W

RB

W

RBW

RBW

S = {(R,R),(R,B),(R,W),(B,R),(B,B),(B,W),(W,R), (W,B),(W,W)}N(S) = 9

R

B

W

B

W

R

B

R

W

Experiment 5: A box containing three pokerchips (one red, one blue, and one white) and two aredrawn without replacement.

S = {(R,B),(R,W),(B,R),(B,W),(W,R), (W,B)} N(S) = 6

Experiment 4-8: A single marble is drawn from abox containing 10 red and 30 blue marbles and its color isrecorded.

S1 = {B,R}

S2 = {R,B,B,B}

EXPERIMENT 4-9: a THUMBTACK ISTOSSED AND THE OUTCOME (WHETHER ITLANDS POINT UP OR POINT DOWN) ISRECORDED.

S ={ },EXPERIMENT A: I choose an attendancecard from today’s “here” pile (randomly)and record the gender of the person towhom it cooresponds.

Experiment B: A card is drawn from astandard deck. You record whether it is aface card or not and what suit it comes from

P(A) = N(A)N(S)

0 < P (A ) < 1

P(A) = 1all outcomes

ONLY TRUEIF ALLPOINTS IN SAREEQUALLYLIKELY

The complement of an event, A is theset of all points in the sample spacethat do not belong to A. This iswritten A (read A complement).

P(A) + P(A) = 1For any event A

How likely is it that two of us have the same birthday?

How many people must be in the room before the probability that two have the same birthday is more than ½?

A little Vocabulary for some common sense ideas:

Two events are Mutually Exclusive if they cannot occur simultaneously. For example in the experiment where you roll a pair of dice, the event that you roll 12 and the event that you roll 2 are mutually exclusive.

If events A and B are Mutually exclusive then the probability that A OR B occurs is P(A) + P(B). Of course the probability that A AND B occurs is 0!

Two events are Independent if knowing that one has occurred does not effect the probability that the other will occur. If I flip a penny and a nickel simultaneously the event that the penny comes up heads and the event that the nickel comes up heads are independent. We assumed that the sex of the children of our alumni friends were independent and that our birthdays were independent.

If A and B are independent the probability that A AND B occur is P(A and B) = P(A)P(B) we used this in the birthday problem

If you have three ribbonsand 4 flowers how many

different ribbon andflower combinations can

you make?

The answer is 12. There are four choicesof flower and three choices of ribbon sothere are 4x3 choices of ribbon flower

pairs. This is an example of theMultiplication Rule.

How manysubsets does ak-element sethave

How many ways can Iassign five tasks tomembers of a group ofseven people without givinganyone more than one task?

How manypermutationsor the lettersMATH arethere?

The Binomial Coefficientn n!k k!(n - k)!

How many kelement subsetsof an n elementset are there?

How many different ways can I choose 3 ofmy 10 closest friends to invite to dinnerFriday night?

A few sample problems from

Ask Marylyn

1 2 31

Suppose 95% of a given population are not drug users and that a drug test has a 95% chance of returning the correct result.

a) What is the probability an individual chosen at random from the given population will test positive for drugs?

b) Given that an individual from this population has tested positive for drugs, what is the probability that the person uses drugs?

c) What is the probability that a non-drug user will test positive?

Binomial Distributions

A Probability Experiment

Probability Quiz

1. What color is my house?a. Blue b. Tan c. Brick

2. What is my brother’s name?a. Michael b. Sean c. Robert

3. What is my Goddaughter’s name?a. Kathleen b. Cassandra c. Heather

4. My neighbor’s name is: a. Jim b. Mark c. Gene

5. My contractor’s name is: a. Bob b. Mark c. Bill

C 5W 4C 4W 3C 4W 3C 3W 2C 4W 3C 3W 2C 3W 2C 2W 1C 4W 3C 3W 2C 3W 2C 2W 1C 3W 2C 2W 1C 2W 1C 1W 0

CW

CW

CW

CW

CW

CW

CW

CW

C

W

C

W

C

W

C

W

C

W

C

W

C

W

n(5) = 1 n(2) =10n(4) = 5 n(1) = 5n(3) =10 n(0) = 1

P(CCCCC) = (1/3)5 = .0041P(CCCCW) = (1/3)4(2/3) = .0082P(CCCWW) = (1/3)3(2/3)2 = .016P(CCWWW) = (1/3)2(2/3)3 = .033P(CWWWW) = (1/3)(2/3)4 = .066P(WWWWW) = (2/3)5 = .132n(5) = 1 n(2) =10n(4) = 5 n(1) = 5n(3) =10 n(0) = 1

P(5) = (1)(.0041) = .0041P(4) = (5)(.0082) = .0412P(3) = (10)(.016) = .1646P(2) = (10)(.0329) = .3292P(1) = (5)(.066) = .3292P(0) = (1) (.132) = .1317

Binomial Probability ExperimentAn experiment that is made up of repeated trials of thesame basic event. The experiment has the followingproperties:

1. Each trial has two possible outcomes (success,failure)2. There are n repeated independent trials3. P(success) = p; P(failure) = q; p + q = 14. The binomial random variable x is the count of thenumber of successful trials that occur. x can take onvalues of 0 to n.

i.e. p= P(success) on an individual trial; q = P(failure)on an individual trial; x = n(successes) for the entireexperiment.

The Binomial Coefficientn n!x x!(n - x)!

Binomial Probability FunctionP(x) = ( )pxq(n-x)n

x

= np

= npq

Mean and Standard Deviation of the Binomial