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5B.1 Permutations and Combinations

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 n P r. Combination An arrangement of r objects from n objects, the

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5B.1 Permutations and Combinations

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  1. 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 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

  2. 5B.1 Permutations and Combinations How many ways are there for a Minnesota Twins manager to 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 = 79,833,600

  3. 5B.1 Permutations and Combinations • How many 4 digit ATM codes are possible • using the digits 0 – 9 if: • the digits cannot be repeated? • the digits can be used more than once? • 10P4 = 10!/(10-4)! = 10!/6! = 5040 or 10*9*8*7 • 10*10*10*10 = 10000

  4. 5B.1 Permutations and Combinations 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

  5. 5B.1 Permutations and Combinations Birthday Problem Twenty people are chosen at random. • What is the probability that none have the same birthday? • What is the probability that at least 2 have the same birthday? Graph y = 1 – (365 nPr x)/(365^x) with a window of [0,47], [0,1]

  6. 5B.1 Permutations and Combinations How many ways are there to make a pizza with toppings of cheese, pepperoni, onions, and sausage if at least one topping is used? • 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

  7. 5B.1 Permutations and Combinations How many ways are there arrange the following letters? 5! = 120 • HALEY • REITER • DEREK 6!/(2!2!) = 180 5!/(2!) = 60

  8. 5B.1 Permutations and Combinations What is the probability of getting four of a kind with 5 cards dealt from a standard deck of 52?

  9. 5B.1 Permutations and Combinations • , • Drake EquationN = 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

  10. 5B.1 Permutations and Combinations http://www.activemind.com/Mysterious/Topics/SETI/drake_equation.html

  11. 5B.2 Permutations of Nondistinct Objects 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)

  12. 5B.2 Permutations of Nondistinct Objects • 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 • actual odds in favor of event A are the reciprocal of the odds against that event, b:a (or ‘b to a’)

  13. 5B.3 Conditional Probability Good Bad 1 2 .405 .045 1 .45 .506 .044 .55 2 Good Bad Good Bad .911 .089 1.00 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? 1.00 .45 .55 .045 .405 .506 .044 P(2| B) = .044/.089 = .49

  14. 5B.3 Conditional Probability Bus Subway bus subway .09 .14 L .23 .21 .56 .77 O late on time late on time .3 .7 1.00 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? 1.00 .30 .70 .21 ..09 .14 .56 P(B| L) = .09/.23 = .39

  15. 5B.4 Probability Trees .95 .0285 .9215 .0485 .05 .0015 .03 .97 1.00 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 probability that the person does, in fact, have the disease? 1.00 .97 .03 .9215 .0285 .0015 .0485 + test says disease - test says no disease

  16. 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.

  17. 5B.5 Bayes Theorem Good Bad Good Bad .931 .09 A .94 acc rej acc rej .019 .041 .06 R .95 1.00 .05 1.00 .05 .95 .019 .009 .931 .041

  18. 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

  19. 5B.5 Bayes Theorem Select a random integer using randInt(1,10) from the calculator. Do not tell the number to anyone. If your integer is 7 or less , then truthfully answer question Q with either a yes or a no. If your number is 8 or greater answer question R with either a yes or a no. Q: Is the last digit in your social security number odd? R: Do you drink?

  20. 5B.5 Bayes Theorem

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