Chapter 6 – The Study of Randomness

• The idea of probability is that chance behavior is unpredictable on the short run, but has a regular pattern in the long run…

• Behavior is random if, while individual outcomes are uncertain, for a large number of repetitions, outcomes are regularly distributed.

• Example: If I roll a die once, I can’t predict with any certainty what number it will land on, but if I roll sixty times, I can expect it to land on 1 ten times, 2 ten times, 3 ten times, etc.

• In the short run, we have no idea what is going to happen with a random event… • BUT, in the long run, a pattern emerges.

• The probability of an outcome is the proportion of times the outcome would occur for a large number of repetitions. (long term relative frequency) • Example: The probability of a die landing on 4 is the proportion of times a die lands on 4 for a large number of repetitions.

• The set of all possible outcomes of an event is the sample space, S , of the event.

• Example: For the event “roll a die and observe what number it lands on” the sample space contains all possible numbers the die could land on.

• S = { 1, 2, 3, 4, 5, 6 }

• An event is an outcome (or a set of outcomes) from a sample space.

• An event is usually denoted by a capital letter. For example, call getting two tails event A.

• The probability of event A is denoted P(A).

• Example 1: When flipping three coins, an event may be getting all heads (HHH). – In this case, the event is one outcome from the sample space. • Example 2: When flipping three coins, an event may be getting two tails. – In this case, the event is a set of outcomes (HTT, TTH, THT) from the sample space.

Event A = Three Heads P(A) = 1 / 8 = 0.125

Event B = Two Tails P(B) = 3 / 8 = 0.375

• The probability of any event is between 0 and 1, inclusive.

0



P(A) 1

• A probability of 0 indicates the event will never occur.

• A probability of 1 indicates the event will always occur.

• If S is the sample space, then P(S) = 1. Some outcome in the sample space will occur.

• The probability that event A does not occur is one minus the probability that A does occur. c

P(A ) 1 P(A)

That A will not occur is called the complement of A and is denoted A

c

.

• Example: When flipping two coins, the probability of getting two heads is 0.25. The probability of not getting two heads is 1 – 0.25 = 0.75.

Event A = Three Heads P(A) = 1 / 8 = 0.125

P(A c )=?

P(A c ) = 1-P(A) P(A c ) = 1-0.125 = 0.875

• If events A and B are disjoint they have no outcomes in common.

A B DISJOINT A NOT DISJOINT B

Example: • Let event A be rolling a die and landing on an even number.

• Let event B be rolling a die and landing on an odd number. • The outcomes for A are {2, 4, 6} • The outcomes for B are {1, 3, 5} • Events A and B are disjoint because they have no outcomes in common. • So the probability of A or B (landing on either an even or an odd number) equals the probability of A plus the probability of B.

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

• Another common term that means the SAME thing as DISJOINT is Mutually Exclusive.

• DISJOINT = MUTUALLY EXCLUSIVE • The terms are interchangeable…

• Events A and B are independent if knowing that one occurs does not change the probability that the other occurs.

Example: • Roll a yellow die and a red die. • Event A is the yellow die landing on an even number • Event B is the red die landing on an odd number. • These two events are independent, because the outcome of A does not change the probability of B.

• If events A and B are independent, then the probability of A and B equals the probability of A multiplied by the probability of B.

P(A and B) = P(A)×P(B)

• Example: The probability than the yellow die lands on an even number and the red die lands on an odd number is:

P(A and B) = P(A)×P(B) 2 1 2 1 4

• If events A and B are independent, –then their complements, A

c

are also independent, –and A

c

–and B

c

is independent of B is independent of A.

and B

c

• What does Mutually Exclusive mean?

• What does Independent mean?

• Venn Diagram Food that students ate.

7

Hamburger

34

Curly Fries

6 45 1 10 12

Hot Dog

25

• How many students are represented?

– 140 • How many students ate curly fries?

– 62 • How many students did not eat a hot dog?

– 92 • How many students ate a Hamburger and Fries?

– 46 • Are Hamburgers and hot dogs disjoint?

– NO!

• What is the probability of randomly selecting a student that had a hot dog and curly fries?

– 11/140 = 0.07857

• Tree Diagram Flipping three coins.

Probabilities Head 0.5

Flip 1 0.5

Tail 0.5

Flip 2 0.5

Head Tail 0.5

Flip 2 0.5

Head Tail Sample Space Flip 3 0.5

0.5

Head Tail Flip 3 0.5

0.5

Head Tail Flip 3 0.5

0.5

Head Tail Flip 3 0.5

0.5

Head Tail

• To determine the number of outcomes in a sample space that has multiple actions – for example, roll 3 dice or flip four coins, • you can find the total number of outcomes by multiplying the individuals.

• roll three dice: 6 outcomes X 6 outcomes X 6 outcomes = 216 total • flip four coins: 2 outcomes X 2 outcomes X 2 outcomes X 2 outcomes = 16 total

• Replacement – when drawing cards, or numbers from a hat, etc. and return the number used.

• Without Replacement – when drawing cards, or numbers from a hat, etc. and you do not return the previously drawn item. This changes the number of total outcomes.

• Example: Draw three cards from a standard deck of cards without replacement. How many outcomes are there?

52 X 51 X 50 = 132,600 total outcomes

Class

Probability

Freshman

0.36

Sophomore

0.25

Junior

0.20

Senior

0.19

• This table is an example of a probability model This model shows the probability of a student being in a certain class if selected at random from a high school.

• What do you notice about the sum of the probabilities?

• Find each of the following…

P(Freshman) P(Senior

c

)

• Equally Likely Outcomes – If a random phenomenon has k possible outcomes, all equally likely, then each individual outcome has probability 1/k.

• Event Probabilities – To find the probability of an event, take the number of outcomes in event A and divide it by the total number of outcomes in the sample space S.

P(A)= count of outcomes in A count of outcomes in S

• OR



–Union of two or more events

• AND



–Intersection of two or more events

P(A or B)

P(A  B)

P(A and B)

P(A  B)

P(A or B)

P(A  B) A B

P(A and B)

P(A  B) A B

• What does Mutually Exclusive mean?

P(A  B) = P(A) + P(B) – P(A  B) A B Special Case: Disjoint

A B

P(A  B) = P(A) + P(B) No overlap means P(A  B) = 0

• P(A) = 0.3

• P(B) = 0.4

• P( A or B ) = 0.65

• Are events A and B mutually exclusive?

• How do you know?

• If events A and B are DISJOINT, then the fact that A occurs tells us that B can not occur… • Disjoint events are NOT Independent

• The first transatlantic telegraph cable was laid in 1866. The first telephone cable across the Atlantic did not appear until 1956 – the barrier was designing “repeaters,” amplifiers needed to boost the signal, that could operate for years on the sea bottom. The first cable had 52 repeaters. The copper cable, laid in 1963 and retired in 1994, had 662 repeaters. The first fiber optic cable, laid in 1988 and has 109 repeaters. There are now more than 400,000 miles of undersea cable, with more being laid every year to handle the flood of internet traffic.

• Repeaters in undersea cables must be very reliable. To see why, suppose that each repeater has probability 0.999 of functioning without failure for 25 years. Repeaters fail independently of each other.

• What is the probability that two repeaters will work for 25 years?

• 10 repeaters?

• 662 repeaters?

• P354 (6.15)

• The probability that event A occurs if we know for certain that event B will occur is called conditional probability. • The conditional probability of A given B is denoted:

P( A | B )

• If events A and B are independent, then knowing that event B will occur does not change the probability of A so for independent events:

P(A | B) = P(A) P(B | A) = P(B)

• Example: When flipping a coin twice, what is the probability of getting heads on the second flip if the first flip was a head?

– Event A: getting head on first flip – Event B: getting head on second flip • Events A and B are independent since the outcome of the first flip does not change the probability of the second flip, so…

P(B | A) = P(B) = 1 2

Venn Diagram: Flip of 2 Fair Coins

0.25

0.25

0.25

P(A) = 0.5

0.25

P(B) = 0.5

P(A  B) = P(A)  P(B|A) A B Special Case: Independent P(A  B) = P(A)  P(B) Knowing A occurred does not effect B so…P(B|A) = P(B)

• Choose a new car or light truck at random and note its color. Here are the probabilities of the most popular colors for vehicles in North America in 2000.

Color

Prob

Silver

0.176

White

0.172

Black

0.113

D Green

0.089

D Blue

0.088

M Red

0.067

Color

Prob

Silver

0.176

White

0.172

Black

0.113

D Green

0.089

• P(color other than listed) = = 1-(0.176+0.172+0.113+0.089+0.088+0.067) = 0.295

• P(Silver or White) = = P(Silver) + P(White) *if disjoint = 0.176 + 0.172 = 0.348

D Blue

0.088

M Red

0.067

Color

Prob

Silver

0.176

White

0.172

Black

0.113

D Green

0.089

D Blue

0.088

• Let S = Silver and W = White • P(S  S) = P(S)P(S) *if independent = (0.176)(0.176) = 0.030976

• P(W  W) = P(W)P(W) *if independent = (0.172)(0.172) = 0.029584

• P((S  S)  (W  W)) = P(S  S)+P(W  W) = 0.0605

M Red

0.067

• The 2000 census allowed each person to choose one or more from a long list of races. That is, in the eyes of the Census Bureau, you belong to whatever race or races you say you belong to. “Hispanic/Latino” is a separate category; Hispanics may be of any race. If we choose a resident of the United States at random, the 2000 census gives these probabilities Asian Black White Other

Hispanic

0.000

0.003

0.060

Non Hispanic

0.036

0.121

0.691

0.062

0.027

• Let A be the event that a randomly chosen American is Hispanic and let B be the event that the person chosen is white.

• P(S) = 1 • P(A) = 0.000+0.003+0.060+0.062 = 0.125

• P(B P(A C )= 1 – (0.060+0.691) = 0.249

• P(Non-Hispanic White) C  B) =0.691

Asian Black White Other

Hispanic

0.000

0.003

0.060

0.062

Non Hispanic

0.036

0.121

0.691

0.027

• The type of medical care a patient receives may vary with the age of the patient. A large study of women who had a breast lump investigated whether or not each woman received a mammogram and a biopsy when the lump was discovered.

Under 65 65 +

YES

0.321

0.365

NO

0.124

0.190

• P(Under65) = 0.321+0.124 = 0.445

• P(65+) = 0.365+0.190 = 0.555

• P(Yes) = 0.321+0.365 = 0.686

• P(No) = 0.124+0.190 = 0.314

Under 65 65 +

YES

0.321

0.365

NO

0.124

0.190

A = 65+ B = YES

• Are A & B Independent?

• P(A  B) ?=? P(A)P(B) From Table P(A  B) = 0.365

Mult Rule P(A) = 0.555 P(B) = 0.686

P(A)P(B) = (0.555)(0.686) = 0.3807

Under 65 65 + • The Values are not the same so they are

NOT

Independent!

YES

0.321

0.365

NO

0.124

0.190

• Deborah and Matthew are anxiously awaiting word on whether they have been made partners of their law firm. Deborah guesses that her probability of making partner is 0.7 and that Matthew's is 0.5. Deborah also guesses that the probability that both she and Matthew are made partners is 0.3.

• P(at least one is promoted) = ?

P(D or M)= P(D) + P(M) – P(D and M) 0.7 + 0.5 - 0.3

= 0.9

D B A C

• Event A = The woman chosen is young (18 to 29) • Event B = The woman chosen is married

Event A = The woman chosen is young (18 to 29) Event B = The woman chosen is married

P(A) =

22, 512 103,870  0.217

P(A and B) =

7,842 103,870  0.075

P(B|A) =

7,842 22,512  0.348

The Multiplication Rule P(A



B) = P(A)



P(B|A)

Only 5% of male high school basketball, baseball, and football players go on to play at the college level. Of these only 1.7% enter major league professional sports.

A = competes in college B = competes professionally

P(A and B)=  0.00085

P(A c and B c )=  0.9499

P(B c |A)= 0.983