5B.1 Permutations and Combinations

Permutation An arrangement of r objects from n objects, the order of which is important. The possible number of such arrangements is denoted by nPr

n Pr n!

(n r)!

Combination An arrangement of r objects from n objects, the order of which is not important. The possible number of such arrangements is denoted by nCr

n C r n!

r!(n r)!

5B.1 Permutations and Combinations

How many ways are there for a Minnesota Twins managerto make out a batting order of 9 players out of a group of 12?

12*11*10*9*8*7*6*5*4 = 79,833,600

or

12 P9 12!

(12 - 9)!

12 *11*10 * 9 * 8* 7 * 6* 5 * 4 * 3* 2 *1

3* 2 *1

= 79,833,600


How many 4 digit ATM codes are possibleusing the digits 0 – 9 if:

a. the digits cannot be repeated?b. the digits can be used more than once?

a. 10P4 = 10!/(10-4)! = 10!/6! = 5040 or 10*9*8*7b. 10*10*10*10 = 10000


How many ways are there to win at Megabucks?Megabucks involves matching 6 numbers out of 54.

54C6 = 54!/6!(54-6)! = 54!/(6!*48!) = 25,827,165


Twenty people are chosen at random.

a. What is the probability that none have the same birthday?b. What is the probability that at least 2 have the same birthday?

.589366

P

366

347 ....

366

364

366

365P(E) a.

2020366

411.2020366c

366

P1P(E) b.

Graph y = 1 – (365 nPr x)/(365^x)with a window of [0,47], [0,1]

Birthday Problem


How many ways are there to make a pizza with toppings ofcheese, pepperoni, onions, and sausage if at least one topping isused?

•With 4 toppings 4C4 = 4!/4!(4-4)! = 1 Note 0! = 1•With 3 toppings 4C3 = 4!/3!(4-3)! = 4•With 2 toppings 4C2 = 4!/2!(4-2)! = 6•With 1 topping 4C1 = 4!/1!(4-1)! = 4

1 + 4 + 6 + 4 = 15 different types of pizzas


How many ways are there arrange the following letters?

•HALEY•REITER•DEREK

5! = 1206!/(2!2!) = 1805!/(2!) = 60


What is the probability of getting four of a kind with 5 cards dealt from a standard deck of 52?

552

14844113

C

CCCkind) a of P(4

2598960

624


5B.1 Permutations and Combinations,

Drake Equation N = R* fp ne fl fi fc L

N = The number of communicative civilizations •R* = The rate of formation of suitable stars (stars such as our Sun) •fp = The fraction of those stars with planets. (Current evidence

indicates that planetary systems may be common for stars like the Sun.) •ne = The number of Earth-like worlds per planetary system

•fl = The fraction of those Earth-like planets where life actually

develops •fi = The fraction of life sites where intelligence develops

•fc = The fraction of communicative planets (those on which

electromagnetic communications technology develops) •L = The "lifetime" of communicating civilizations


http://www.activemind.com/Mysterious/Topics/SETI/drake_equation.html

The payoff odds against event A represent the ratio of net profit (if you

win) to the amount of the bet.

Payoff odds against event A =

(net profit):(amount bet)

5B.2 Permutations of Nondistinct Objects

actual odds in favor of event A are the reciprocal of the odds against that event, b:a (or ‘b to a’)

actual odds against event A occurring are the ratio P(A)c/ P(A), usually

expressed in the form of a:b (or ‘a to b’), where a and b are integers with no common factors

5B.2 Permutations of Nondistinct Objects

A manufacturer has two machines that produce a certain product. Machine 1 produces 45% of the product and Machine 2 produces 55% of the product. Machine 1 produces 10% defective items and Machine 2 produces 8% defective items. If a defective item is produced, what is the probability it was produced by Machine 2?

1 2

GoodGood Bad Bad

1.00

.45 .55

.405 .045 .506 .044

.044.506

.045.405

Good Bad

1

2

.45

.55

.911 .089 1.00

P(2| B) = .044/.089 = .49

5B.3 Conditional Probability

A man takes a bus or a subway to work with probabilities .3 and.7 respectively. When he takes the bus, he is late 30% of the days.When he takes the subway, he is late 20% of the days.If he is late, what is the probability he took the bus?

bus subway

latelate on time on time

1.00

.30 .70

..09 .21 .14 .56

.56.21

.14.09

Bus Subway

L

O

.23

.77

.3 .7 1.00

P(B| L) =.09/.23 = .39

5B.3 Conditional Probability

.03 .97

1.00

.0285 .0015 .0485 .9215

.03 .97

.95

.05

1.00

.9215.0285

.0015 .0485

.0285.0285 .0485

.37

A blood test for a certain disease is 95% accurate and 3% of the population has the disease. A person is chosen at random and blood test indicates that they have the disease. What is the probabilitythat the person does, in fact, have the disease?

5B.4 Probability Trees

+ test says disease- test says no disease

Suppose we know that a food inspector accepts 98 % of all good shipments and has incorrectly rejected 2 % of all good shipments. In addition, the inspector accepts 94% of all shipments, and it is known that 5% of all shipments are of inferior quality.

a. Find the probability that a shipment is rejected.b. Find the probability that a shipment is good.c. Find the probability that a shipment is good and accepted.d. Find the probability that a shipment is of inferior quality and accepted.e. Find the probability that a shipment is accepted, given that it is of inferior quality.f. Find the probability that a shipment is rejected, given that it is good.

5B.5 Bayes Theorem

Good Bad

accacc rej rej

1.00

.95 .05

.931 .019 .009 .041

.041.019

.09.931

Good Bad

A

R

.94

.06

.95 .05 1.00

5B.5 Bayes Theorem

Suppose we know that a food inspector accepts 98 % of all good shipments and has incorrectly rejected 2 % of all good shipments. In addition, the inspector accepts 94% of all shipments, and it is known that 5% of all shipments are of inferior quality.

a. Find the probability that a shipment is rejected.b. Find the probability that a shipment is good.c. Find the probability that a shipment is good and accepted.d. Find the probability that a shipment is of inferior quality and accepted.e. Find the probability that a shipment is accepted, given that it is of inferior quality.f. Find the probability that a shipment is rejected, given that it is good.

.06.95

.931.009

.009/.05 = .18

.019/.95 =.02

5B.5 Bayes Theorem

5B.5 Bayes Theorem

Select a random integer using randInt(1,10) fromthe calculator. Do not tell the number to anyone.If your integer is 7 or less , then truthfully answerquestion Q with either a yes or a no. If your number is 8 or greater answer questionR with either a yes or a no.

Q: Is the last digit in your social security number odd?R: Do you drink?

5B.5 Bayes Theorem

YES NO

SSN ODD 0 0

DRINK 0 0

0

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5B.1 Permutations and Combinations