Probabilities, combinitorics, permutations
Quiz yourself by thinking what should be in
each of the black spaces below before clicking
on it to display the answer.
Help!
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| P(E) | # of ways E can occur
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total possible outcomes
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| Sample Space | collection of all possible outcomes
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| Event | a collection of outcomes
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| outcome | any ONE thing that can happen
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| in a regular 52 card deck what is the probability of getting an ace? | 4
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52
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| Empirical Probability | Counting observation, changes by observation
# of times E occurred/ total # of observations
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| Contingency table | make a grid to figure out the probabilities
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| 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
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| Make a list | STAT ENTER
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| _ X | Sample Mean
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| SX | sample standard deviation
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| OX | Population Standard Deviation
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| n | Sample Size
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| Med | Median (50% mark)
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| To compute 2 lists of descriptive stats | STAT > ENTER enter data in list > STAT > CALC > > ENTER > 2ND > 1,2 (etc - use comma)
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| 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?
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| Combination = NCR | What is the probability of getting X from A any order?
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| Combination - What is the probability of getting a 5 card hand, all spades. | 13 spades 13 c 5 cards
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52 c 5 cards = 4.95 x 10-4
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| Permutation - In a 52 card deck how many 5 card hands are there? | 52 c 5 = (math PRB NCR enter 5 enter)
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| n factorial | The number of way to arrange n objects
n MATH PRB ! ENTER ENTER
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| 52 card deck arrange 7 cards on table, how many ways | 52 P 7 = 52 MATH PRB NPR ENTER (52 npr) 7 ENTER
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| 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
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| Complimentary Probability | Given E, its opposite E is its compliment
P(E) + P(E!)=1 so, P(E) = 1-P(E!)
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| 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)
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| 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."
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| 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
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| Independence (2 events) | P(E given F) = P(E)
Independent if proportion are the same
dependent of proportions are not the same.
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| 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
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| 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
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| 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])
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| 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
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| 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
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| 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
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| 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
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| 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)
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| 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
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| 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
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| 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
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| 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)
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| 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)
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| 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
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| 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
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| Empirical Rule: Standard Normal Distribution |
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onesmartgirl
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