APSTAT SECTION IV PROBABILITY

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APSTAT SECTION IVPROBABILITY
2
CHAPTER 14From Justin To Kelly(or from
randomness to probability)
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Randomenss and Probability Basics

• What is Random?
• Individual outcomes are unpredictable in the
  short run
• In the long run, however, outcomes are regular
  AND predictable (LOLN - Law of Large Numbers)
• Lets do a simulation
• Flip 10, 100, 1000, 10000 coins
• Find of heads to nearest whole
• sum(randint(0,1,10))

4
Random Simulation

• Short Run Long Run
• 10 Flips 100 Flips 999Flips

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What is Probability

• Over a HUGE number of trials (probability is
  Long-Term), the proportion of times an outcome
  would occur.
• Typically expressed by P and a range from 0 to 1
• 0 being never ever happens
• 1 being always happens
• We can only ESTIMATE real-world probabilities
• Can be expressed as a , but not as cool.

6
Models of Probability

• Two Main Thangs
• LIST all possible outcomes
• ASSIGN a probability to each outcome
• ie. Year in school probability _at_ WPS

FROSH SOPH JUNIOR SENIOR
60/230 60/230 70/230
40/230
P .261 .261 .304 .164
Should add up to 1.0, but may be a bit off due to
ROUNDING ERROR
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Vocab Time

• Sample Space S Set of all possible outcomes
• Event Any outcome or set of outcomes
• ie. Freshman
• ie. Juniors AND seniors

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CRAPS! Roll Them Bones!

• Disclaimer Gambling can be dangerous and
  addictive, plus over the LONG RUN, the casino
  always wins. So dont gamble, buy a casino!
• Sample space when 2 die are rolled

36 potential outcomes
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CRAPS! Roll Them Bones!

• Event Rolling a 7 when pips are added
• ProbSpeak P(Roll a 7)

P(Roll 7) 6/36 .167
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CRAPS! Roll Them Bones!

• Event Rolling a 8 when pips are added
• ProbSpeak P(8)

P(8) 5/36 .139
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CRAPS! Roll Them Bones!

• Event Rolling a Hard 8 (two 4s)
• ProbSpeak P(Hard 8)

P(8) 1/36 .028
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More Sample Space

• Same problem can have different look at sample
  space
• If in craps, if all we care about are pips
• S (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)

P(2) 1/36 .028
P(3) 2/36 .056
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Key Points -

• Independence
• One outcome does not affect another outcome
• ie. If I roll a 6 with one die, it wont affect
  my chances of rolling a 6 with the second die
• WITH Replacement
• Example pick a card from a deck, put it back and
  pick anotherP(2 Aces)
• WITHOUT Replacement
• Example pick two cards from a deck without
  replacing the first cardP(2 Aces)

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Rules o Probability

• Probability of an event is between 0 and 1
• PROBSPEAK
• All possible probabilities add up to 1
• PROBSPEAK
• Probability of an event NOT happening is 1 minus
  the probability of the event happening. The
  probability of an event NOT happening is called
  the COMPLIMENT of an event and is written Ac
• PROBSPEAK

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Rules o Probability Continued

• If two events are disjoint (mutually exclusive),
  they have no outcomes in common. For example, in
  craps, rolling a 5 AND a 7 is disjoint, one roll
  cant produce both outcomes.
• Therefore (for disjoint events)
• AND (for disjoint events)..

S
A
B
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Rules o Probability Continued Again

• If two events are NOT disjoint (not mutually
  exclusive) but ARE independent . For example,
  roll 2 dice
• Event A Die 1 Shows a 6 P(A)1/6
• Event B Die 2 Shows a 6 P(B)1/6
• P(A and B) P(A)P(B)
• 1/6 1/6 1/36 .028ish

S
A
AB
B
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Disjoint Events Are NOT Independent

• Hurting your brain?
• Just thinkIf I roll two die and add up the pips,
  what are the chances that I get a 5 and a 7.
• Thats why (in disjoint
• events)
• P(A and B)0

S
Roll 5
Roll 7
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CHAPTER 15Probability Goes Crazy with the Cheez
Whiz
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Tree Diagrams
OUTCOMES

• ie. Flip 3 Coins, Count of Heads

3H, 0T
H
H
2H, 1T
T
H
H
2H, 1T
T
T
1H, 2T
H
2H, 1T
H
T
1H, 2T
T
H
1H, 2T
T
FLIP 1
T
0H, 3T
FLIP 2
FLIP 3
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3 Flip outcomes
3 Heads 2 Heads 1 Head 0 Heads
1/8 3/8 3/8
1/8
P .125 .375 .375 .125
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More (and evilererer) Probability Rules

• Addition Rule 3 or more disjoint events

S
A
C
B
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Addition Rule Non-Disjoint Events

• Find P(A or B)
• If I do P(A)P(B) the area AB gets counted twice
• To make it work, I do P(A) P(B) and then
  subtract one P(AB)

S
A
AB
B
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Example P(Male Or Senior Citizen)

• P(Male)0.5
• P(65)0.2
• Assume independence for Maleness Oldness

S
A
AB
B
.5 .2 -
(.5)(.2) .7 - .1
.6
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Joint Events

• Not always independent - Cant assume
• Example Survey of music tastes at WPS
• Probability of student liking hip-hop (A)
• P(A)0.5
• Probability of student liking rap (B)
• P(B)0.4
• Think! Isnt there a decent chance that people
  who like hip hop may be more likely to like rap
  as well.
• Proportion of students who like BOTH rap and hip
  hop
• P(A and B)0.3

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Joint Events - Continued

• Probability of student liking hip-hop (A)
• P(A)0.5
• Probability of student liking rap (B)
• P(B)0.4
• Proportion of students who like BOTH rap and hip
  hop
• P(A and B)0.3

S
HipHop
BOTH
0.1
0.2
0.3
RAP
0.4
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Joint Events Same thing Using a table

• Probability of student liking hip-hop (A)
• P(A)0.5
• Probability of student liking rap (B)
• P(B)0.4
• Proportion of students who like BOTH rap and hip
  hop
• P(A and B)0.3

Rap
0.2
Hip-Hop
0.4
0.5
0.1
0.6
S
HipHop
BOTH
0.1
0.2
0.3
RAP
0.4
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Conditional Probability

• Main Idea
• Probability can change if we know some other
  event has occurred
• World Poker Championships
• Flushes are good All same suit
• You get 2 cards that are secret, then 5 cards
  are dealt for the community
• You make the best 5-card hand you can

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World Poker Championships

• My Hand Community Cards
• ? ? ? ? ? ?
• Wow, Im close to a flush! What is the
  probability that the last card (the river) is a
  ??
• Overall, the chance of a ? is 13/52 or .25, but
  we already know what 6 cards are and that 4 of
  them are ?s
• Find Probability(? given that 4 of 6 visible
  cards are ?s)
• ProbSpeak P(Spade 4 of 6 visible spades)
• Think

?
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General Rule for Any Two Events

• P (A and B) P(A)P(B?A)
• Example
• Probability of getting 2 aces in two successive
  draws (no replacement)
• P(Ace on 1st and Ace on 2nd)
• P(Ace on 1st)P(Ace on 2nd ? Ace on 1st)
• 4/523/510.0045
• Notice if replacement (independence), the formula
  still works since P(Ace on 2nd ? Ace on 1st)
  P(Ace on 1st)
• Therefore 4/524/520.0059

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Definition for Conditional Probability

• P (A and B) P(A)P(B?A)
• Take this old Formula and solve for P(B?A)

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Using Decision Trees

• The following information gives information on
  DVD players sold by a certain electronics store
• Let B1 Event that Brand 1 is purchased
• Let B2 Event that Brand 2 is purchased
• Let E Event that Warranty is purchased
• Therefore P(B1).7 P(B2).3
• AND!!!!!! P(E ? B1).2 P(E ? B2).4

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Using Decision Trees - Continued

• P(B1).7 P(B2).3 P(E ? B1).2 P(E ?
  B2).4

(.7)(.2).14
(.7)(.8).56
(.3)(.4).12
(.3)(.6).18
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Using Decision Trees Continued 2

• NOW ANSWER QUESTIONS!!!!

What proportion of DVD purchasers also purchased
the warranty? P(B1 and E) P(B2 and
E) P(E).14.12.26
(.7)(.2).14
(.7)(.8).56
(.3)(.4).12
(.3)(.6).18
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Using Decision Trees Bayess Rule

• What is probability of B1 given E

What proportion of DVD purchasers also purchased
the warranty?
(.7)(.2).14
P(B1 and E) .14 P(E).14.12.26
(.7)(.8).56
P(B1 ? E) .14/.26
P(B1 ? E) 0.539
(.3)(.4).12
(.3)(.6).18
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CHAPTER 16Random Variables
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Discrete Random Variables

• Discrete????
• Just means that there are a reasonable
  (countable) number of options.
• What do we do with them? List outcomes and then
  probabilities
• Answer questions
• Easy as pie

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Example Rolling 2 dice

• Find P(Xgt9) P(10)P(11)P(12).083.055.027
  .165

• Find P(X?9) P(9)P(10)P(11)P(12)
• .121.083.055.027 .286

• Find P(5ltXlt8) P(6)P(7).139.167 .316

Really bad Histogram Thanks Microsoft!
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Continuous Random Variables

• Continuous?
• Not countable
• Example, think of all possible decimals between 0
  and 1Boy thats a lot!
• If we threw down a histogram of a gazillion
  random numbers between 0 and 1, wed get this

Density Curve Area underneath is 1.0
Uniform Distribution
1.0
0.0
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Find Probabilities (Just area of rectangle)

• P(Xgt0.8)

1.0
0.8
0.0
0.2
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Find Probabilities (Just area of rectangle)

• P(.2ltXlt0.8)

1.0
0.8
0.2
0.0
0.6
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Check this out!

• What is the probability P(X0.8)?

1.0
0.8
0.0
0! ZERO! ZIP! NADA!
Why? Area of a straight line is Zero, Yeah?
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SO

• P(Xgt0.8) is the same as P(X?0.8)
• With Continuous Variables, It does not matter
  which one you use
• Cool, Huh???

1.0
0.8
0.0
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Whats the sassiest density curve?

• NORMAL DISTRIBUTION. YEAH!!!

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Male Height N(68,2)

• Let XHt in Inches
• Find P(Xgt71)
• Normalcdf(71,100000000,68,2)
• P(Xgt71)_____

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Means and Variances of Random Variables

• Example 2005 AP Stat Scores

Remember x-bar is a sample mean, but we are
talking about the entire population of AP Stat
test takers, so we must use m (population mean).
mx 1(.13)2(.23)3(.25)4(.19)5(.20) 3.1
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Variances of Random Variables

• Recall Variance is (Standard Deviation)2
• Here is the formula, It looks icky, but its
  pretty easy to use

Outcome Value
Sum
Variance of X
Mean of outcomes
Outcome Probability
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Means and Variances of Random Variables
mx 3.1
1.7903
Standard deviation would be the square root of
this. 1.31ish
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Law of Large Numbers

• How can we find the actual m of mens heights?
• Not really realistic to measure every man in the
  world
• Use x-bar as a reasonable estimate
• Gets more reasonable as the sample size increases
  Thats the LAW OF LARGE NUMBERS.
• The larger the sample size, the more likely x-bar
  will approach the m.

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Rules for Means

• If I taught in Canada, they would not dig the
  average height of males in inches, they like
  centimeters. Plus, all men there measure their
  heights while wearing 8cm high pumps. Very
  stylish!
• How does that change the mean???

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Rules for Means

• Here is the rule
• Here is what we do with those Canadian heights
  (1in?2.54cm)

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Rules for Means 2

• In volleyball there are two main blocking
  statistics, solo blocks (by self) and assisted
  blocks (with a buddy). If Chrissa Trudelle
  averaged .5 solo blocks and 1.3 assisted blocks
  per match, how many total blocks did she average?
• Rule
• DO IT!

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Rules For Variances

• OK, the last rule was ridiculously easy, but this
  next stuff is a bit rough.
• Think about the mens height and the changes in
  Canada with the centimeters and 8cm pumps.
• How would these change the variance?

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Rules For Variances

• If we add the same value (8cm) to every height,
  how does variance (and standard deviation) change?

• Right, variance does not change if I add the same
  value to each height

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Rules For Variances

• If we multiply each observation by the same
  amount what will that do?

• Right, multiplying the variance by a factor will
  change the variance. Greater if gt1 or if lt-1.
  Less if between 1 and -1

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Rules For Variances Linear Transform

• If given N(68,2) for average male height, and we
  transform it again with 82.5X, what happens to
  the variance?
• Rule
• DO IT!

Standard deviation would be the square root of
this. 5
56
Rules For Variances Add/Subtract

• Here are the formulas

p (rho) is like r, it shows the correlation
between X and Y and is between -1 and 1. Should
be stated unless X and Y are independent
Dont these look familiar???
57
Check this out

• Rearrange the formulas a bit..

Perfect square trinomials???
58
Speaking of Rho

• That little p only affects things if there is
  some correlation between the variables
• If the problem lists a rho, you gotta use it
• If it doesnt list a rho, but it should have, do
  the problem without it, but talk about how there
  could be some correlation which would affect the
  variance (or standard deviation)
• If no correlation p 0. Therefore

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Lets use it now!

• Coach Boff sweeps the gym floor in N(10,2)
  minutes and mops the floor in N(15,3) minutes.
  Assume that the time sweeping and mopping are
  independent. Find the mean and standard
  deviation of the combined time.
• Mean is easy. 1015 25 minutes
• Now lets find the standard deviation

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Lets use it now!
REMEMBER! You can not add standard deviations,
you must square them to get variances, add the
variances and then square root the sum!
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But if X and Y had p .5
Why more? If rho is positive, X more likely to
be higher if Y is also higher. Variation moving
in the same way will increase the variance.
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Now you try!

• Mr. Riebhoff and Mr. Marsheck are the nations
  1030th best partner biathlon team. Mr. Riebhoff
  will do the running leg which is a 10k road race
  where he has historically had a time of N(48,5)
  in minutes. Marsheck will do the bike ride of
  50k where he has historically had a time of
  N(106, 10) in minutes. What are the mean and
  standard deviation of their combined finish times?

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Remember

• Show formula(ae) first
• Talk about any assumptions you are making
• Dont forget that standard deviation is the
  square root of variance
• Have fun!

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CHAPTER 17Probability Models
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Binomial Distribution

• 4 Requirements for a Binomial Distribution
• 2 outcomes Success/Failure
• I.e. Heads or Tails, Boy or Girl Baby, Make or
  Miss a Shot
• Independent observations
• Probability does not change when you learn the
  result of a previous event
• Probability for success (p) is constant for all
  observations
• FIXED NUMBER OF OBSERVATIONS!!!!!!!!!
• 5 Free throws, 17 exam questions, 20 Students

THE KEY
66
Important parts

• n of Observations
• Fixed for a binomial distribution
• p Probability of success
• Defined by you or the question
• x of successes
• Can be from 0 to n

67
Do you remember

• Normal Distribution
• N(m,s) ex. N(68,2)
• NOW! Binomial Distribution
• B(n,p)
• Example
• A 70 free throw shooter shoots
• 10 Free throws
• B(10,0.7)

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Which of these would be Binomial?

• Flip a fair coin and count number of flips until
  a head appears.
• 350 students at WPS. 10 are 6th graders.
  Choose 10 names at random with no replacement and
  count of 6th graders.
• Shaq is a 52 free throw shooter. Observe next
  10 free throws and count of makes.

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Binomial PDF

• Remember Normal CDF?
• NOW Binomial PDF

Cumulative Distribution Function
Probability Distribution Function
70
Binomial PDF 10 FT _at_ 70

• B(10,0.7)
• X 0 to 10

.3
.2
.1
0 1 2 3 4
5 6 7 8 9
10
Binompdf(10,0.7,0)
Binompdf(10,0.7,1)
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Cumulative Distribution Function

• Cumulative
• It accumulates, adds up
• EXAMPLE 70 FT Shooter, 10 FTs

X
PDF
CDF
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Graph the CDF
1.0
.75
.50
.25
0 1 2 3 4
5 6 7 8 9
10
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Formulae for Binomial Distribution

• Mean For Binomial Distribution
• m np
• Makes sense yeah?
• Example, I flip a coin 16 times, how many heads?
• m 16(.5) 8

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Formulae for Binomial Distribution

• Standard Deviation For Binomial Distribution
• s
• Why? Sausage. Just deal and know where it is on
  the Formulae Sheet.
• Ex. Find SD of 10 FT Problem
• s
• s 1.449

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LETS DO IT!

• Find Mean and Standard Deviation on 20 Free
  Throws if
• p0.7
• p0.8
• p0.9
• p0.99
• What happens to m as p approaches 1.0?

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Math Attack

• Remember Factorials? -- n!
• Examples
• 5! 54321 120
• 3! 321 6
• Now the crazy stuff
• 0! 1
• Kinda Like a0 1, yah?
• Well need these in a minute, youll see why.

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Binomial Coefficient

• of ways I can get k successes in n tries.
• Example How many ways can I get three tails in
  5 flips?
• Old Skool Way (easy to mess up)
• TTTHH TTHTH TTHHT THTTH
• THTHT THHTT HTHTT HTTTH
• HTTHT HHTTT

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Impress your friends at the next math party way

• Pascals Triangle
• 1
• 1 1
• 1 2 1
• 1 3 3 1
• 1 4 6 4 1
• 1 5 10 10 5 1

5 Choose 0
5 Choose 1
5 Choose 2
5 Choose 3
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Formula Way

• Formula
• Use it! 5 Choose 3

n choose k
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Binomial Probability

• Recall 3 coins flipped, X of Heads

X 0 1 2 3 1
3 3 1
P(X0) 1P(HC)3 .125
P(X1) 3P(HC)2 P(H) .375
P(X2) 3P(H)2 P(HC) .375
P(X3) 1P(H)3 .125
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Imagine doing P(5 heads in 9 flips)

• What we need is a formula

Insert binomial coefficient here
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Lets do 3 flips P(2 heads)
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Now You Try!

• In a previous chapter, we found that the
  probability of rolling a 7 (craps!) with two fair
  die is 0.167. Let X be the number of 7s rolled
  in a series of 10 rolls

84
Now You Try!

• 1 Find the probability that 3 7s will be
  rolled in the 10 attempts

85
Now You Try!

• 2 Use your TI-83 and find the distribution of
  X
• binompdf(trials,p,x)
• 0 1 2 3 4 5 6 7 8
  9 10

86
Now You Try!

• 3 Find the m and s of the number of 7s that
  would be rolled in 10 attempts
• m
• s

87
Geometric Distribution

• 4 Requirements for a Geometric Distribution
• 2 outcomes Success/Failure
• I.e. Heads or Tails, Boy or Girl Baby, Make or
  Miss a Shot
• Independent observations
• Probability does not change when you learn the
  result of a previous event
• Probability for success (p) is constant for all
  observations
• Looking for of trials needed for 1
  success!!!!!!!
• Flip a coin, how many flips until 1st Head?

THE KEY
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Geometric vs. Binomial

• Binomial
• Shoot 10 FTs with p(make)0.7 find p(8 makes)
• Geometric
• With p(make) 0.7, Shoot until 1st make, count
  the number of attempts

89
Identify the Geometric Distributions

• A Flip a coin until you get a head
• B Record the number of times a player makes
  both shots in a one-and-one foul-shooting
  situation. (In this situation, you get to
  attempt a second shot only if you make the first)
• C Draw a card from the deck, observe it and
  replace it into the deck. Count the number of
  times you draw a card in this manner until you
  observe a jack.

90
Identify the Geometric Distributions

• D Buy a pick 6 lottery ticket every week
  until you win the lottery. Count the of weeks
  it takes for you to win.
• E There are 10 red marbles and 5 blue marbles
  in a jar. You reach in, and without looking,
  select a marble. You want to know how many
  marbles you need to draw (without replacement),
  on average, in order to be sure that you have 3
  red marbles.

91
CRAPS! Roll till a 7 shows up
FORMULA FOR GEOMETRIC PROBABILITIES
92
Lets try it!

• Mr. Riebhoff is U-G-L-Y (he aint got no alibi).
  In college, he had only a 20 chance of a
  randomly selected woman (he used a random
  table) agreeing to meet him for a soda.

93
Lets try it!

• 1 Find a probability distribution from x
  1 to x 5 that shows x the of females he
  would ask before getting a yes
• 1 2 3 4 5

94
Lets try it!

• 2 Make a CDF of the data from 1
• 1 2 3 4 5

1.0
0.0
95
Lets try it!

• 3 What is the probability that after 5 girls
  asked, Riebhoff would still be dateless?

96
Using the TI-83

• geometpdf(p,x)
• In Riebhoff Date Example
• geometpdf(0.2,1)
• geometpdf(0.2,2)
• geometpdf(0.2,3)
• geometpdf(0.2,4)

Number if trials till success
97
Mean of Geometric Random Variable

• Common Sense
• Guess what the mean number of rolls I would need
  to roll a 5 on a fair die?
• Guess the mean number of flips I would need to
  get a head on a fair coin?
• m 1/p