1. 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 order of which is not important. The possible number of such arrangements is denoted by n C r 5B.1 Permutations and Combinations

2. 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 5B.1 Permutations and Combinations

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

4. How many ways are there to win at Megabucks? Megabucks involves matching 6 numbers out of 54. 54 C 6 = 54!/6!(54-6)! = 54!/(6!*48!) = 25,827,165 5B.1 Permutations and Combinations

5. 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? Graph y = 1 – (365 n P r x)/(365^x) with a window of [0,47], [0,1] Birthday Problem 5B.1 Permutations and Combinations

6. 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 4 C 4 = 4!/4!(4-4)! = 1 Note 0! = 1 With 3 toppings 4 C 3 = 4!/3!(4-3)! = 4 With 2 toppings 4 C 2 = 4!/2!(4-2)! = 6 With 1 topping 4 C 1 = 4!/1!(4-1)! = 4 1 + 4 + 6 + 4 = 15 different types of pizzas 5B.1 Permutations and Combinations

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

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

9. , Drake Equation N = R* f p n e f l f i f c L N = The number of communicative civilizations R* = The rate of formation of suitable stars (stars such as our Sun) f p = The fraction of those stars with planets. (Current evidence indicates that planetary systems may be common for stars like the Sun.) n e = The number of Earth-like worlds per planetary system f l = The fraction of those Earth-like planets where life actually develops f i = The fraction of life sites where intelligence develops f c = 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. 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

12.  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

13. 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? 12 Good Bad 1.00.45.55.405.045.506.044.506.045.405 Good Bad 1 2.45.55.911.089 1.00 P(2| B) =.044/.089 =.49 5B.3 Conditional Probability

14. 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? bussubway late on time 1.00.30.70..09.21.14.56.21.14.09 Bus Subway L O.23.77.3.7 1.00 P(B| L) =.09/.23 =.39 5B.3 Conditional Probability

15. .03.97 1.00.0285.0015.0485.9215.03.97.95.05 1.00.9215.0285.0015.0485 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? 5B.4 Probability Trees + test says disease - test says no disease

16. 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

17. GoodBad acc rej 1.00.95.05.931.019.009.041.019.09. 931 Good Bad A R.94.06.95.05 1.00 5B.5 Bayes Theorem

18. 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

19. 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