Chapter 8: Probability: The Mathematics of Chance · 2013. 11. 14. · A continuous probability...

Chapter 8: Probability: The Mathematics ofChance

November 11, 2013

Chapter 8: Probability: The Mathematics of Chance

Last Time

Probability Models and Rules

Discrete Probability Models

Equally Likely Outcomes


Probability Rules

Probability Rules

Rule 1. The probability P(A) of any event A satisfies 0 ≤ P(A) ≤ 1.

Rule 2. If S is the sample space in a probability model, then P(S) = 1.

Rule 3. The complement rule: P(AC ) = 1 − P(A).

Rule 4. The multiplication rule for independent events:P(A and B) = P(A) × P(B).

Rule 5. The general addition rule:P(A or B) = P(A) + P(B) − P(A and B).

Rule 6. The addition rule for disjoint events:P(A or B) = P(A) + P(B).


Counting distinct items

Counting Ordered Collections of Distinct Items

Rule A. Suppose we have a collection of n distinct items. Wewant to arrange k of these items in order, and the same item canappear more than once in the arrangement. The number ofpossible arrangements is

n × n × · · · × n = nk

Rule B. (Permutations) Suppose we have a collection of ndistinct items. We want to arrange k of these items in order, andany item can appear no more than once in the arrangement. Thenumber of possible arrangements is

n × (n − 1) × · · · × (n − k + 1)


Counting Distinct Items

Counting Unordered Collections of Distinct Items

Rule C. Suppose that we have a collection of n distinct items. Wewant to select k of those items with no regard to order, and anyitem can appear more than once in the collection. The number ofpossible collections is

(n + k − 1)!

k!(n − 1)!

Rule D. (Combinations) Suppose that we have a collection of ndistinct items. We want to select k of these items with no regardto order, and any item can appear no more than once in thecollection. The number of possible selections is

n!

k!(n − k)!


Question

In poker, a royal flush is a 5-card hand containing an ace, king,queen, jack, and 10, all of the same suit.

(a) How many royal flush hands are possible?Answer:

4

(b) What is the number of 5-card hands possible from a 52-carddeck? Answer: (

52

5

)(c) What is the probability that 5 cards drawn at random from a

52-card deck will yield a royal flush? Answer:

4(525

)Chapter 8: Probability: The Mathematics of Chance

Question

A computer assigns three-character log-in IDs that may containthe digits 0 to 9 as well as the letters a to z , with repeats allowed.

1 What is the probability that your ID contains no x?Answer: (

35

36

)(35

36

)(35

36

)2 What is the probability that your ID contains no digits?(

26

36

)(26

36

)(26

36

)3 What is the probability that your ID contains exactly 2 x ’s?

3

362


Question

What is the probability that two people in this room have the samebirthday?

Answer:

1 −365!

(365−14)!

36514= 0.223


Question

What is the probability that two people in this room have the samebirthday?Answer:

1 −365!

(365−14)!

36514= 0.223


Question

How many different ways can you seat 10 couples at a circulartable such that everyone is sitting next to their spouse?

2(9!)

Assuming we alternate men and women how arrangements haveone couple not sitting together?

8(9!)


This time

Continuous Probability Models


Question

If I let you choose a number between 0 and 1, what is theprobability that you will chose a number between .2 and .4?

P(.2 ≤ X ≤ .4) = 0.2



Density Curve

A density curve is a curve that

is always on or above the horizontal axis and

has area exactly 1 underneath it.


A continuous probability model is a probability model thatassign probabilities as areas under a density curve. The area underthe curve and above any interval of values is the probability of anoutcome in that interval.


Question

Would you rather flip a coin for $1 or roll a dice for $30,000?


Mean of Probability Model

Mean of a Discrete Probability Model

Step 1: Make a table with two rows. The first row needs to list all thepossible numerical outcome values in the sample space.

Step 2: In the second row of the table, list the respective probabilitiesof each of the outcome values from the first row of the table.

Step 3: Write a third row where each entry is the product of the twoitems in the same column from the first two rows. Now addup all the values in the third row, and you will get the meanof the discrete probability model.


Law of Large Numbers

Law of Large Numbers

Observe any random phenomenon having numerical outcomes withfinite mean µ. According to the law of large numbers, as thephenomenon is repeated a large number of times,

the proportion of trials in which an outcome occurs gets closerand closer to the probability of that outcome, and

the mean x̄ of the observed values gets closer and closer to µ.


Standard Deviation

Standard Deviation of a Discrete Probability Model

Suppose that the possible outcome x1, x2, · · · , xk in a samplespace S are numbers, and that pj is the probability of outcome xj .The standard deviation of a discrete probability model withmean µ is denoted by the lowercase Greek letter sigma (σ) and isgiven by this formula:

σ =√

(x1 − µ)2p1 + (x2 − µ)2p) + · · · + (xk − µ)2pk


Question

If I pay you the amount you roll on a dice, what is the mean andthe Standard Deviation?

µ =1

6(1 + 2 + 3 + 4 + 5 + 6) =

21

6=

7

2= 3.5

σ =

√1

6((1 − 3.5)2 + (2 − 3.5)2 + (3 − 3.5)2 + (4 − 3.5)2 + (5 − 3.5)2 + (6 − 3.5)2) = 1.70783


Question

Question 1: Should you buy the extended warranty on a newwashing machine? Suppose there are two outcomes an 85 percentprobability of needing no repairs, and a 15 percent probability ofneeding no repairs, and a 15 percent probability of needing a $ 200repairs, and a 15 percent probability of needing a $200 repairduring the warranty period. Based on the mean outcome for thismodel, what would be a “break-even” price to you for the extendwarranty?Question 2: If you have to pay $ 1 to play a hand of black jack,how much money would you have to win to make it worth playing?


Question

An American roulette wheel has 38 slots numbered 0,00, and 1 to36. The ball is equally likely to come to rest in any of these slotswhen the wheel is spun. The slot numbers are laid out on a boardon which gamblers place their bets. One column of numbers onthe board contains multiples of 3. Joe places a $1 “column bet”that pay out $ 3 if any of these numbers comes up.

1 What is the probability model for the outcome of one bet,taking into account the $1 cost of a bet?

2 What are the mean and standard deviation for this model?

3 Joe plays roulette every day for years. What does the law oflarge numbers tell us about this results?


Next Time

Central Limit Theorem


Chapter 8: Probability: The Mathematics of Chance · 2013. 11. 14. · A continuous probability...

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