Birthday Problem• What is the smallest number of

people you need in a group so that the probability of 2 or more people having the same birthday is greater than 1/2?

• Answer: 23No. of people 23 30 40 60Probability .507.706.891.994

Probability

•Formal study of uncertainty•The engine that drives Statistics

• Primary objective of lecture unit 4:

1. use the rules of probability to calculate appropriate measures of uncertainty.

2. Learn the probability basics so that we can do Statistical Inference

Introduction• Nothing in life is certain• We gauge the chances of

successful outcomes in business, medicine, weather, and other everyday situations such as the lottery or the birthday problem

Tomorrow's Weather

A phenomenon is random if individual

outcomes are uncertain, but there is

nonetheless a regular distribution of

outcomes in a large number of repetitions.

Randomness and probabilityRandomness ≠ chaos

Coin toss The result of any single coin toss is

random. But the result over many tosses

is predictable, as long as the trials are

independent (i.e., the outcome of a new

coin flip is not influenced by the result of

the previous flip).

The result of any single coin toss is

random. But the result over many tosses

is predictable, as long as the trials are

independent (i.e., the outcome of a new

coin flip is not influenced by the result of

the previous flip).

First series of tossesSecond series

The probability of heads is 0.5 = the proportion of times you get heads in many repeated trials.

4.1 The Laws of Probability

1. Relative frequencyevent probability = x/n, where x=# of occurrences of event of interest, n=total # of observations– Coin, die tossing; nuclear power plants?

• Limitationsrepeated observations not practical

Approaches to Probability

Approaches to Probability (cont.)

2. Subjective probabilityindividual assigns prob. based on personal experience, anecdotal evidence, etc.

3. Classical approachevery possible outcome has equal probability (more later)

Basic Definitions

• Experiment: act or process that leads to a single outcome that cannot be predicted with certainty

• Examples:1. Toss a coin2. Draw 1 card from a standard deck of

cards3. Arrival time of flight from Atlanta to

RDU

Basic Definitions (cont.)

• Sample space: all possible outcomes of an experiment. Denoted by S

• Event: any subset of the sample space S;typically denoted A, B, C, etc.Null event: the empty set Certain event: S

Examples1. Toss a coin once

S = {H, T}; A = {H}, B = {T}2. Toss a die once; count dots on upper

faceS = {1, 2, 3, 4, 5, 6}A=even # of dots on upper face={2, 4, 6}B=3 or fewer dots on upper face={1, 2, 3}

3.Select 1 card from adeck of 52 cards.S = {all 52 cards}

Laws of Probability

1)(,0)(.2

event any for ,1)(0 1.

SPP

AAP

Coin Toss Example: S = {Head, Tail}Probability of heads = 0.5Probability of tails = 0.5

3) The complement of any event A is the event that A does not occur, written as A.

The complement rule states that the probability

of an event not occurring is 1 minus the

probability that is does occur.

P(not A) = P(A) = 1 − P(A)

Tail = not Tail = Head

P(Tail ) = 1 − P(Tail) = 0.5

Probability rules (cont’d)

Venn diagram:

Sample space made up of an event

A and its complement A , i.e.,

everything that is not A.

Birthday Problem• What is the smallest number of

people you need in a group so that the probability of 2 or more people having the same birthday is greater than 1/2?

• Answer: 23No. of people 23 30 40 60Probability .507.706.891.994

Example: Birthday Problem

• A={at least 2 people in the group have a common birthday}

• A’ = {no one has common birthday}

502.498.1)'(1)(

498.365

343

365

363

365

364)'(

:23365

363

365

364)'(:3

APAPso

AP

people

APpeople

Unions: , orIntersections: , and

A

A

Mutually Exclusive (Disjoint) Events

• Mutually exclusive ordisjoint events-no outcomesfrom S in common

A and B disjoint: A B=

A and B not disjoint

A

A

Venn Diagrams

Addition Rule for Disjoint Events

4. If A and B are disjoint events, then

P(A or B) = P(A) + P(B)

Laws of Probability (cont.)

General Addition Rule

5. For any two events A and B

P(A or B) = P(A) + P(B) – P(A and B)

19

For any two events A and B

P(A or B) = P(A) + P(B) - P(A and B)P(A or B) = P(A) + P(B) - P(A and B)

A

B

P(A) =6/13

P(B) =5/13

P(A and B) =3/13

A or B

+_

P(A or B) = 8/13

General Addition Rule

Laws of Probability: Summary

• 1. 0 P(A) 1 for any event A• 2. P() = 0, P(S) = 1• 3. P(A’) = 1 – P(A)• 4. If A and B are disjoint events, then

P(A or B) = P(A) + P(B)• 5. For any two events A and B,

P(A or B) = P(A) + P(B) – P(A and B)

M&M candies

Color Brown Red Yellow Green Orange Blue

Probability 0.3 0.2 0.2 0.1 0.1 ?

If you draw an M&M candy at random from a bag, the candy will have one

of six colors. The probability of drawing each color depends on the proportions

manufactured, as described here:

What is the probability that an M&M chosen at random is blue?

What is the probability that a random M&M is any of red, yellow, or orange?

S = {brown, red, yellow, green, orange, blue}

P(S) = P(brown) + P(red) + P(yellow) + P(green) + P(orange) + P(blue) = 1 P(blue) = 1 – [P(brown) + P(red) + P(yellow) + P(green) + P(orange)]

= 1 – [0.3 + 0.2 + 0.2 + 0.1 + 0.1] = 0.1

P(red or yellow or orange) = P(red) + P(yellow) + P(orange)

= 0.2 + 0.2 + 0.1 = 0.5

Example: toss a fair die once

S = {1, 2, 3, 4, 5, 6}• A = even # appears = {2, 4, 6}• B = 3 or fewer = {1, 2, 3}• P(A orB) = P(A) + P(B) - P(A andB)

=P({2, 4, 6}) + P({1, 2, 3}) - P({2})

= 3/6 + 3/6 - 1/6 = 5/6

End of First Part of Section 4.1