II - Probability Counting Techniques three rules of counting 1multiplication rules 2permutations 3combinations Section 2 - Probability (1)
II - Probability Counting Techniques 1multiplication rules In a sequence of n events in which the first one has k 1 possibilities, the second has k 2, etc., the total possibilities of the sequence will be: (k 1 )(k 2 )(k 3 )...(k n ) If the number of possibilities in each of the events is the same (k), the solution is: k n Section 2 - Probability (2)
II - Probability Counting Techniques 1multiplication rules eg, roll a die twice... how many possiblities? eg, building a wood deck... can choose from - 2 types of fasteners - 4 species - 3 finishes How many decks are possible? Section 2 - Probability (3)
II - Probability Counting Techniques 1multiplication rules tree diagrams: Section 2 - Probability (4)
II - Probability Counting Techniques 2permutations an arrangement of n objects in a specific order eg, 3 cars in a 3 car garage: - Mustang - Cougar - Nova How many ways can you park them? Section 2 - Probability (5)
II - Probability Counting Techniques 2permutations eg, 3 cars in a 3 car garage (cont.): number of permutations: (n)(n-1)(n-2)... n! Section 2 - Probability (6)
II - Probability Counting Techniques 2permutations Permutation Rule #1: The number of permutations of n objects all taken together is: P = n n n! Section 2 - Probability (7)
II - Probability Counting Techniques 2permutations eg, 5 cars in a 3 car garage: - Mustang - GTO - Cougar - Camaro - Nova How many ways can you park them? Section 2 - Probability (8)
II - Probability Counting Techniques 2permutations Permutation Rule #1: The number of permutations of n objects all taken together is: P = n n n! Permutation Rule #2: The arrangement of n objects in a specific order taking r objects at a time is: n P r = n! ( n r)! Section 2 - Probability (9)
II - Probability Counting Techniques 2permutations eg, 8 cars (indistinguishable by make): - 3 Mustangs - 2 Cougars - 3 Novas How many ways can you arrange them in an 8 car garage if you cannot tell the difference between alike car types? Section 2 - Probability (10)
II - Probability Counting Techniques 2permutations Permutation Rule #1: The number of permutations of n objects all taken together is: P = n n Permutation Rule #2: The arrangement of n objects in a specific order taking r objects at a time is: Permutation Rule #3: n! P r n! ( n r)! The arrangement of n objects in which k 1 are alike and k 2 are alike, etc. is: n P = = k! k n!! K k 1 2 n! Section 2 - Probability (11)
II - Probability Counting Techniques 3combinations a selection of objects without regard to order eg, 5 cars: - Mustang - GTO - Cougar - Camaro - Nova In how many ways can you choose 3 cars (how many combinations of 3?) Section 2 - Probability (12)
II - Probability Counting Techniques 3combinations eg, 5 cars (cont.): Section 2 - Probability (13)
II - Probability Counting Techniques 3combinations Combination Rule: The number of combinations of r objects selected from n objects is: n C r = ( n n! r)! r! Section 2 - Probability (14)
II - Probability Counting Techniques Review: chances of winning at Lotto 649: How many ways can you arrange letters of the alphabet taking 3 at a time? Section 2 - Probability (15)
II - Probability Counting Techniques Review: How many hands of 5 cards are possible? What is the probability of drawing a full house with 2 Aces and 3 Jacks? Section 2 - Probability (16)
Probability Distributions game: draw an ace win $10 draw a face card win $5 draw anything else lose $3 Section 2 - Probability (17)
Probability Distributions random variables variables with associated probabilities values are determined by chance probability distribution consists of the values that a random variable can assume and their corresponding probabilities (determined theoretically or by observation) probabilities are assigned to each possible outcome and results are evaluated Section 2 - Probability (18)
Probability Distributions eg 2 coin experiment: S = (HH, HT, TH, TT) assume x = # heads probability distribution: Section 2 - Probability (19)
Probability Distributions continuous probability distribution: f ( x) = e ( x µ ) 2 2 σ 2π σ 2 Section 2 - Probability (20)
Probability Distributions 2 requirements of probability distributions: ( X ) P = 1 0 P( X ) 1 eg roll a dice: let x = the outcomes X 1 2 3 4 5 6 P(X) 1/6 1/6 1/6 1/6 1/6 1/6 Section 2 - Probability (21)
Probability Distributions X 1 2 3 4 5 6 P(X) 1/6 1/6 1/6 1/6 1/6 1/6 - the mean for this probability distribution? - what is the average outcome? Section 2 - Probability (22)
Probability Distributions generalized formula: µ ( ) ( ) ( ) = X P X + X P X + K+ 1 1 2 2 X P n X n µ = n i= 1 X P ( ) i X i Section 2 - Probability (23)
Probability Distributions eg 2 coin experiment: let x = the number of Heads X 0 1 2 P(X) 1/4 1/2 1/4 Section 2 - Probability (24)
Probability Distributions expectation: the expected value of a random variable, E(X), in a probability distribution is the theoretical (long-run) average of the variable... µ = E n ( X ) = X ( ) i P X i i= 1 Section 2 - Probability (25)
Probability Distributions game: draw an ace win $10 draw a face card win $5 draw anything else lose $3 Section 2 - Probability (26)
Probability Distributions variance & standard deviation: the theoretical spread of the distribution of the random variable σ 2 = n [( ) ( )] 2 X i µ P X i i= 1 working formula n 2 σ = i X i µ i= 1 [ ( )] 2 2 X P Section 2 - Probability (27)
Probability Distributions X 1 2 3 4 5 6 P(X) 1/6 1/6 1/6 1/6 1/6 1/6 - the variance and standard deviation of the probability distribution for rolling a dice... Section 2 - Probability (28)
Probability Distributions eg a community forestry committee consists of 5 members, up to 2 of which can be representatives from an environmental NGO... Historical evidence shows that NGO reps have sat on this committee with the following probabilities... X = # of NGO reps 0 1 2 P(X) 1/10 6/10 3/10 How many NGO reps do we expect, on average (in the long run)? What is the theoretical standard deviation of this distribution? Section 2 - Probability (29)
Probability Distributions eg (cont.) X = # of NGO reps 0 1 2 P(X) 1/10 6/10 3/10 Section 2 - Probability (30)
1uniform distributions 2binomial distributions 3multinomial distributions 4Poisson distributions 5geometric / negative binomial distributions 6hypergeometric distributions Section 2 - Probability (31)
1uniform distributions random variable, X, assumes the values X 1, X 2, X 3,..., X n ; all with equal probabilties eg, throw a dice... the probability of any outcome is 1 / n where n = the number of outcomes Section 2 - Probability (32)
Section 2 - Probability (33) III - Probability 1uniform distributions ) ( 1 1 = = = = n i i i n i i X P X n X µ ( ) n X n i i = = 1 2 2 µ σ
2binomial distributions used when: an experiment consists of repeated trials (n trials) each trial has only two outcomes (success / failure) the outcome of each trial is independent the probabilities of successes / failures are the same for the duration of the experiment Section 2 - Probability (34)
2binomial distributions ie, a trial is repeated n times... the probability of a success is p the probability of a failure is q or 1 - p the random variable, X, is the number of success that you are interested in P n ( ) X X X p q n = X Section 2 - Probability (35)
2binomial distributions eg, the probability of insect infestation is 0.3 n = 20 trees p = 0.30 q = 1-0.30 = 0.70 what is the probability that 4 trees (out of 20) are infested? Section 2 - Probability (36)
2binomial distributions eg, insect infestation (cont.) Section 2 - Probability (37)
2binomial distributions eg, the probability of insect infestation is 0.3 n = 20 trees p = 0.30 q = 1-0.30 = 0.70 what is the probability less than 4 trees (out of 20) are infested? Section 2 - Probability (38)
2binomial distributions eg, insect infestation (cont.) Section 2 - Probability (39)
2binomial distributions mean and variance (standard deviation) of binomial distributions: µ = np σ 2 = npq = np 1 ( p) σ = σ 2 Section 2 - Probability (40)
2binomial distributions eg, the probability of insect infestation is 0.3 n = 20 trees p = 0.30 q = 1-0.30 = 0.70 what is the mean and variance of this binomial distribution? Section 2 - Probability (41)
2binomial distributions eg, insect infestation (cont.) Section 2 - Probability (42)
3multinomial distributions P like binomial, but used when an experiment can result in more than 2 outcomes (k outcomes) reverts back to binomial when there are only 2 outcomes ( ) X1 X 2 X k X = p p K p where: X n 1, X 2, L, X k X is the number of times each event occurs ΣX = n (total outcomes) P s are the respective probabilities 1 2 k Section 2 - Probability (43)
3multinomial distributions eg, lumber is graded at a grading station... P (graded correctly) = 0.95 P ( misgraded) = 0.04 P (missed) = 0.01 If we ran 20 boards through, what is the probability that......13 are graded correctly...5 are misgraded...2 are missed Section 2 - Probability (44)
3multinomial distributions eg, grading station (cont.) Section 2 - Probability (45)
4Poisson distributions like binomial, but n is very large and p is small generally used when a sample yields a random variable that is: the number of outcomes in a given time interval eg, number of plane crashes per year densities in a given volume or area eg, number of trees per hectare assumes outcomes are independent of each other and that probability of occurrence is small and constant Section 2 - Probability (46)
4Poisson distributions P ( X ) = e µ µ X! X where: µ = the average number of items that occur in a given time or space interval X = the number of items that we are concerned with e = the natural log, 2.7183 Section 2 - Probability (47)
4Poisson distributions eg, If a sales office receives, on average, 1 phone call per hour, what is the probability that it receives (exactly) 3 in one hour? Section 2 - Probability (48)
4Poisson distributions eg, If a sales office receives, on average, 1 phone call per hour, what is the probability that it receives less than 3 in one hour? note: µ = σ 2 Section 2 - Probability (49)
5 geometric / negative binomial distributions geometric distributions: same as binomial, but the question is what is the probability of the first success on the X th trial? P ( X ) = p q X 1 where: p = probability of success q = probability of failure X = trial of interest Section 2 - Probability (50)
5 geometric / negative binomial distributions eg, If you flip a coin repeatedly, what is the probability of getting your first Head on the 5 th flip? Section 2 - Probability (51)
5 geometric / negative binomial distributions negative binomial distributions: same as binomial, but the question is what is the probability of the K th success on the X th trial? P X ( ) k k X p q X = k 1 1 where: p = probability of success q = probability of failure X = trial of interest k refers to the k th success Section 2 - Probability (52)
5 geometric / negative binomial distributions eg, If you flip a coin repeatedly, what is the probability of getting your second Head on the 5 th flip? Section 2 - Probability (53)
5 geometric / negative binomial distributions Note that negative binomial reverts back to geometric if k = 1 eg, If you flip a coin repeatedly, what is the probability of getting your first Head on the 5 th flip? Section 2 - Probability (54)
5 geometric / negative binomial distributions eg, If you roll a die repeatedly, what is the probability of getting your second 3 on the 10 th roll? Section 2 - Probability (55)
6hypergeometric distributions like binomial, but sampling is done without replacement in other words, trials are not independent -- the outcome of one affects the probability of another (p changes) problem is most prevalent in smaller samples, where one outcome has a noticeable affect on subsequent probabilities Section 2 - Probability (56)
6hypergeometric distributions - derive by way of example... eg, If you draw 3 cards from a deck (without replacement), what is the probability that they are all Spades? tree diagram: Section 2 - Probability (57)
6hypergeometric distributions eg, 3 Spade example (cont.) classical probability: Section 2 - Probability (58)
6hypergeometric distributions eg, 3 Spade example (cont.) classical probability: Section 2 - Probability (59)
6hypergeometric distributions where: P ( X ) = k X N N n N = number in population k = # of items labelled a success n = number in sample X = random variable (number of success that we are interested in) n k x Section 2 - Probability (60)
6hypergeometric distributions eg, 3 Spade example (cont.) hypergeometric distribution: Section 2 - Probability (61)
6hypergeometric distributions eg, 3 Spade example (cont.) hypergeometric distribution: Section 2 - Probability (62)
6hypergeometric distributions eg, Furniture components are sold to a manufacturer... - packages contain 292 boards - on average, 14 boards per package are defective The manufacturer, before buying, will sample 20 boards at random (without replacement). If more than 1 defect is found, the manufacturer will reject the package. What is the probability that a package will be rejected? Section 2 - Probability (63)
6hypergeometric distributions eg, furniture components example (cont.) Section 2 - Probability (64)
6hypergeometric distributions eg, furniture components example (cont.) Section 2 - Probability (65)
6hypergeometric distributions eg, furniture components example (cont.) Section 2 - Probability (66)
6hypergeometric distributions mean and variance (standard deviation) of hypergeometric distributions: µ = ( ) n k N σ 2 = ( n) 1 k N k N σ = σ 2 Section 2 - Probability (67)
6hypergeometric distributions Note: when N is very large relative to n, k / n approaches p (constant), and hypergeometric distribution reverts back to binomial... " p " = k N Section 2 - Probability (68)
6hypergeometric distributions Rule of Thumb: in a hypergeometric distribution, use binomial when... n N < 0.05 Section 2 - Probability (69)