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Statistics 2

Probabilities, combinitorics, permutations

P(E) # of ways E can occur ______________________ total possible outcomes
Sample Space collection of all possible outcomes
Event a collection of outcomes
outcome any ONE thing that can happen
in a regular 52 card deck what is the probability of getting an ace? 4 ___ 52
Empirical Probability Counting observation, changes by observation # of times E occurred/ total # of observations
Contingency table make a grid to figure out the probabilities
The Law of Large Numbers (difference gets smaller w/ more observations) as you increase the number of observations the probability increases. P(|Pemp(E)-Pcl(E)|>S) -> 0
Make a list STAT ENTER
_ X Sample Mean
SX sample standard deviation
OX Population Standard Deviation
n Sample Size
Med Median (50% mark)
To compute 2 lists of descriptive stats STAT > ENTER enter data in list > STAT > CALC > > ENTER > 2ND > 1,2 (etc - use comma)
Permutation = NPR the number of ways to arrange r objects from a larger set of n objects. (one at a time) in a 52 card deck, how many ways to arrange?
Combination = NCR What is the probability of getting X from A any order?
Combination - What is the probability of getting a 5 card hand, all spades. 13 spades 13 c 5 cards ____________ 52 c 5 cards = 4.95 x 10-4
Permutation - In a 52 card deck how many 5 card hands are there? 52 c 5 = (math PRB NCR enter 5 enter)
n factorial The number of way to arrange n objects n MATH PRB ! ENTER ENTER
52 card deck arrange 7 cards on table, how many ways 52 P 7 = 52 MATH PRB NPR ENTER (52 npr) 7 ENTER
How many ways can you choose 5 people from a class of 50? 50 p 5 50 MATH PRB NPR ENTER (50 NPR) 5 ENTER
Complimentary Probability Given E, its opposite E is its compliment P(E) + P(E!)=1 so, P(E) = 1-P(E!)
Complimentary Prob - in a 52 card deck what is probability of getting one ace in 5 card hand? opposite of getting one ace is no ace = 1 - 48 C 5 / 52 C 5 (48 cards = 52 - aces - 1 ace in your hand)
Conditional probability Given 2 events (E given F) "Given a 52 card deck, what is the probability getting 5 spades, given all cards are black."
Given a 52 card deck, what is the probability getting 5 spades, given all cards are black P (5 spades/ black) #ways to get 5 spades # ways black 5 card hands spades 13 C 5 / 26 (black cards) C 5 = .0195
Independence (2 events) P(E given F) = P(E) Independent if proportion are the same dependent of proportions are not the same.
Are aces independent of hearts? P (ace/hearts) vs P(aces/cards) ace in hearts/# hearts = # aces /total # cards 1/13 = 4/52 (1/13) independent
Binomial Distribution (Bernouli) Roulette = Bernouli Poker blackjack = not bernouli Finite trials, 2 events (success vs fail), P(success) same from trial to trial, trials are independent
The probability of getting the flu is 18% Out of 10 friends, what is the probability of: between 2 and 4 get the flu P(X<n)=binom cdf (n [# of trials], p [probability of success],n [number of success])
What is the probability of sick at c- restaurant is 15%, out of 5 visits that you'll get sick twice? Note exact # (Binom pdf) 2nd VERS BINOMPDF (5,.15, 2) visits possible, %, sick actual
What is the probability of sick at c- restaurant is 15%, out of 5 visits that you'll get sick AT LEAST twice? Note range (Binom cdf) 1 - BINOMCDF (5, .15, 2) opposite of 2 or more - visits poss, %, top range - 1
Geometric Distribution: Basketball player has 54% free throw average. What is the probability he misses 4 and makes the 5th Note: Probability of 1st success on nth try 2nd VARS GEOMETCDF (.54, 5) (never pdf) probability, tries
If the probabilty of getting a role in a movie is 2.1%, what is the probability of getting it for the first time on the 1th try? 2nd VARS GEOMETPDF (.021, 10) = .0173
What is the probability of winning the lottery in your working lifetime: play once a week for 30 years 52 x 30 = 1560 times (3 steps incl. prob) 1/50 nCr 5 X 45 nCr 1 1 #/50 pos #s nCr 5# X 45 remaining # nCr 1 last # = 1.048 X 10-8 FINAL STEP: 1 - binompdf (1560, 1.05x10-8)
Given that the probability of being accepted by an ivy league school is 4.2%, what is the probability of: Getting in after 3 rejections? Specific # but no order GeometPDF (.042, 4) [after 3 means 4] .00369
Given that the probability of being accepted by an ivy league school is 4.2%, what is the probability of: getting in if you apply 4 times Exact # BinomPDF (4, .042, 1) = .1477 tries, %, success
The Poisson Distribution computing averages given data over time is used when: potentially infinite # of trials Trials are independent you have an average # of success per trial 4 errors/sec 5 strokes/ hole
Max safety level for ameb dysentary is 6 live org per liter. what is probability you get 7 live org? P(X) = Poisson PFD Exact P(<X) = Poisson CDF Range (7-1 = 6) 1 - PoissonCDF (6,6) = .3937 average,(7-1=6)
An internet service provider can handle up to 32 customers/sec what is the probability that any 5 sec intreval has at least 145 customers/ sec? Average # success - Poisson dist. 32 cust/sec x 5 sec = 160 AVERAGE MAX 1 - POISSONCDF (160, 144) = .8911 average, X (145 - 1 + 144)
Fatal auto accident rate in a suburb is 1/4500/year If the suburb has a population of 180,000 what is the probability of no more than 2 fatal accidents in a given year? 1/45,000 into 180,000 = 4 POISSONCDF (4,2) = .23810 average,X
Is the range open ended? (either end) Asking for probability? Draw out Graph, Mean is the center point NormalCDF continuous-U or L parameter is missing (10 99,L,M,O) (U,10 99,M,O) NORMALCDF (U, L, M, O)requires upper & lower Upper, Lower, Mean, Standard Deviation
Empirical Rule: Standard Normal Distribution
Created by: onesmartgirl on 2010-04-04

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