Question | Answer |
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 | |