In how many ways can the letters of SEA be arranged? In how many ways can the letters of SEE be arranged?

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3 -Pick up Quiz Review Handout by door -Turn to Packet p. 5-6 In how many ways can the letters of SEA be arranged? In how many ways can the letters of SEE be arranged? - Take Out Yesterday s Notes we ll finish a few notes first today! - Get out warm-up paper What s the difference? How do you account for it?

4 In how many ways can the letters of SEA be arranged? In how many ways can the letters of SEE be arranged? What s the difference? How do you account for it?

5 The number of n objects of which p are alike and q are alike is: n is total number of objects p is the amount of one alike object number of multiple e s or 7 s, etc q is the amount of another alike object Ex: How many arrangements of SEE are possible? n! pq!!

6 How many different 11-letter patterns can be formed with letters of MISSISSIPPI? 11! 4!4!2! 34,650 How many different 9-letter patterns can be formed with letters of TENNESSEE? 9! 4!2!2! 3780 How many different 7-letter patterns can be formed with the letters of WYOMING? Why is this third 7! 5040 example different? The letters of Wyoming are all distinct.

7 Recall or events mean addition because all of these could be possibilities: option1 + option2 + option3 However and events means multiply because all of these will occur together: option1 * option2 * option3

8 n C r n! n! ( n r)! r! n r! r! 3. A local restaurant is offering a 3 item lunch special. If you can choose 3 or fewer items from a total of 7 choices, how many possible combinations can you select? C C C C A hockey team consists of ten offensive players, seven defensive players, and three goaltenders. In how many ways can the coach select a starting line up of three offensive players, two defensive players, and one goaltender? C C C

9 The members of a string quartet composed of 2 violinists, a violist, and a cellist are to be selected from a group of 6 violinists, 3 violists, and 2 cellists, respectively. a) In how many ways could the string quartet be formed? b) In how many ways can the string quartet be formed if one of the violinists is to be designated as 1 st violinists and the other is to be designated as 2 nd violinists?

10 Day 5 First 8 in 8 video BDTc6ME&feature=youtu.be&hd=1 We ll watch this while we practice

11 Day 4 Warm-up: Permutations vs. Combinations 1. If you have a standard deck of cards in how many different hands exists of: a) 5 cards b) 2 cards 2. Choose 3 desserts from a menu of 8 desserts 3. Choose a winner and a runner up from the 40 Miss Pickle Princess contestants 4. How many different 11-letter arrangements are there for a) PALINDROMES b) PRICELESSLY? 5. Assign the part of a play to the 4 different lead characters from a group of 20 who tried out and 3 backstage crew members (they all have the same job) from a group of 5. Riddle: What integer between when spelled out is in alphabetical order?

12 Warm-up: Permutations vs. Combinations 1. If you have a standard deck of cards in how many different hands exists of: a) 5 cards b) 2 cards 52C5 2,598,960 52C2 1, Choose 3 desserts from a menu of 8 desserts Combination 8C 3 = Choose a winner and a runner up from the 40 Miss Pickle Princess contestants Permutation 40P 2 = How many different 11-letter arrangements are there for a) PALINDROMES b) PRICELESSLY? 11! = 39,916,800 11! (2!2!2!) 4,989, Assign the part of a play to the 4 different lead characters from a group of 20 who tried out and 3 backstage crew members (they all have the same job) from a group of 5. Perm&Comb 20 P 4 * 5 C 3 = 1,162,800

13 Riddle: What integer between when spelled out is in alphabetical order? Forty!

14 This is an unusual paragraph. I'm curious as to just how quickly you can find out what is so unusual about it. It looks so ordinary and plain that you would think nothing was wrong with it. In fact, nothing is wrong with it! It is highly unusual though. Study it and think about it, but you still may not find anything odd. But if you work at it a bit, you might find out. Try to do so without any coaching!

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16 Homework Day 4 Tonight s HW = Quiz Review Sheet & Study for tomorrow s Quiz Study your notations, Formulas, etc!! Check Review Sheet answers online tonight!

17 Section 7.1

18 An Experiment is an activity with observable results. (called outcomes) Sample Space: The set of all possible outcomes Must use S = {,,, } **must include ALL outcomes! Event: subset of a sample space List events using {...}, {...},

19 Ex. Rolling a die Outcomes: landing with a 1,2,3,4,5, or 6 face up Sample Space: S = {1,2,3,4,5,6} Events:, {1}, {2}, {3}, {4}, {5}, {6}, S S is that certain event (contains all outcomes) Like the Universal set so it must occur is an impossible event. (no elements or outcomes)

20 Examples a. Tossing a coin b. Choosing a card from a deck of cards c. Drawing a marble from a bag containing two red and three blue S={Heads, Tails} S={R, R, B, B, B} W

21 An experiment consists of spinning the hand on the disk below twice. If it lands on a line, spin again. Find the sample space. Then determine the event E in which at least one B occurs. Sample Space: B P G S={ BB, GB, BG, BP, PB, GG, GP, PG, PP} Event E: {BB}, {GB}, {BG}, {PB}, {BP}

22 Let S = {q, r, t} be a sample space of an experiment. List all of the events of this experiment., {q}, {r}, {t}, {q, r}, {q, t}, {r, t}, {q, r, t} or (S) When asked to write ALL events, include empty set and S. *Similar to #23 in your HW

23 You have gone to the SPCA to adopt a puppy. You would like a poodle or cocker spaniel, that is brown or grey, and has either a red or orange collar. How many possible puppies fit your criteria? List the sample space. S={PBR}, {PBO}, {PGR}, {PGO}, {CBR}, {CBO}, {CGR}, {CGO}

24 The union of events A & B is the event A B The intersection of events A & B is the event A B The complement of event A is the event A c

25 Review Example: Rolling a die. S={1,2,3,4,5,6} Let A = rolling a number less than 4 B = rolling an odd number Find: A B = {1,2,3,5} A A B B c = {1,3} = {2}

26 Let P be any sample space and W, R, and S be any three events. Describe the given events using symbolic notation. 1. The event that S and W occur. 2. The event that R and S do not occur. 3. The event that W or R occur and not S. 4. Given events W and S, only one of the two occurs. c c S W ( R S) c or R c S c ( W R) S ( W S ) ( W S) c

27 Complementary events are two outcomes of an event that are the only two possible outcomes. Ex: Complementary: Flipping a coin and getting heads or tails. Ex: Not Complementary: Rolling a die and getting a 1 or 2 All complementary events are mutually exclusive, but all mutually exclusive events are not necessarily complementary.

28 Events A & B are mutually exclusive if A B Mutually Exclusive Events (Disjoint Events): Two or more events that cannot occur at the same time. Describe two events that are mutually exclusive. Are these?? Rolling an even # and rolling an odd # on a die Yes- Mutually Exclusive Drawing a single card from a deck of cards and having it be a diamond and a red card. No! They can occur at the same time so they are NOT Mutually Exclusive.

29 YOU TRY! Are the events mutually exclusive? Find the probability. Spinner with numbers 1-8 1) What is the probability of spinning a 4 and a 6 at the same time on a single spin. Mutually Exclusive so the probability is 0. 2) Spinning an even number and a multiple of 3 at the same time on a single spin. NOT Mutually Exclusive (could be 6) so the probability is 1/8 3) Spinning an even number and a prime number on a single spin. Not Mutually Exclusive (could be 2) so probability is 1/8. 4) Spinning an even number and a number less than 2 on a single spin. Mutually Exclusive so the probability is 0.

30 An experiment consists of tossing a coin 3 times and observing the resulting sequence of heads and tails. Find the sample space of the experiment. (Hint: you may need to draw a tree diagram) > answer on next slide Determine the event E that exactly two heads appear. Determine the event F that at least one head appears.

31 An experiment consists of tossing a coin 3 times and observing the resulting sequence of heads and tails. Determine the event E that exactly two heads appear. {HHT}, {HTH}, {THH} Determine the event F that at least one head appears. Remember, Events are Sets -> use Set notation! {HHH}, {HHT}, {HTH}, {HTT}, {THH}, {THT}, {TTH}

32 An experiment consists of casting a pair of dice and observing the number that falls uppermost on each die. Create the sample space S for this experiment. (Hint: Create a table or chart) Die (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) 2 (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) 3 (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) 4 (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) 5 (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) 6 (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) Determine the events E 3 and E 7 that the sum of the numbers is 3 or 7, respectively. {1, 2}, {2, 1}, {3, 4}, {4, 3}, {2, 5}, {5, 2}, {6, 1}, {1, 6} Remember, Events are Sets -> use Set notation! *Needed for #21 in your HW

33 PRACTICE An experiment consists of casting a pair of dice and observing the number that falls uppermost on each die. Die (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) 2 (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) 3 (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) 4 (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) 5 (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) 6 (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) a) What are the events of rolling a product less than 6? {1, 1}, {2, 1}, {3, 1}, {4, 1}, {5, 1}, {1, 2}, {2, 2}, {1, 3}, {1, 4}, {1, 5} b) What are the events of rolling an odd number on the first die and a 4 on the second die? {1, 4}, {3, 4}, {5, 4}

34 Think of an experiment. Make it interesting. Don t use anything we ve discussed. Describe the sample space of the experiment. Construct two events, E and F, of the experiment. Find the union and intersection of E and F and the complement of E. Are E and F mutually exclusive? Explain. We ll share these with the rest of the class. (FUN!)

35 Homework Day 4 Tonight s HW = Quiz Review Sheet & Study for tomorrow s Quiz Study your notations, Formulas, etc!! Check Review Sheet answers online tonight!

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