Finite Mathematics MAT 4: Chapter 8 Notes Counting Principles; More David J. Gisch The Multiplication Principle; Permutations Multiplication Principle Multiplication Principle You can think of the multiplication principle as counting the branches (combinations) on a tree diagram.
Multiplication Principle Example: You have 2 pairs of shoes, 5 pairs of pants and 6 different tops. How many outfits can you make? Multiplication Principle Example: A housing builder provides the following options: Deck or patio. Hardwood, carpet or linoleum Granite, quartz, or compound Expensive landscaping, cheap landscaping or no landscaping. How many different combinations do they offer? Multiplication Principle Example: How many possible 0 digit phone numbers can there be if the first and fourth digit cannot be a zero? Multiplication Principle (Dependence) Example: For the Iowa Lottery one must choose 5 out of 59 numbers for the white balls; then choose out of 35 numbers for the Powerball. How many possible tickets are there? 2
Multiplication Principle (Dependence) Example: You have 6 books and want to place them on a shelf. In how many ways can you organize the books? Factorial! As you saw in the last few examples we often end up multiply successive terms until we reach one. There is a mathematical notation for this. Factorial! Example: 5 people on a committee need to be lined up for a photo. In how many ways can this be done? Factorial! Example: 42 cars are in a race. How many possible finishing orders can there be? 3
Permutations Permutations Let s consider the last example. What if we only cared about the possible top 5 finishing orders? Go to math and arrow over to PRB. Permutations Example: You have 6 books and want to place 4 of them on a shelf. In how many ways can you organize the books? Permutations Example: There are 5 people on a committee. How many ways can they select a president, vice president, and treasurer? 4
MAT 4 - Chapter 8 Permutations Example 8..0: A television talk show will include 4 men and 5 women as panelists. (a) In how many ways can the panelists be seated? (b) In many ways can they be seated if they need to alternate? Repetition in Permutations Suppose you are asked to arrange some objects but the objects are indistinguishable. For example, arranging the numbers, 2, 2,,. You cannot tell the ones apart or the twos apart. If you have to arrange objects where of them repeat, of them repeat, of them repeat, Then you can calculate the number of arrangements as!!!! (c) In how many ways can they be seated if the men and women must be seated together? Permutations (Repetitions) Example: In how many ways can you arrange the letters of the word Mississippi? Permutations (Repetitions) Recall when were taking marbles out of the bag and I kept saying, in that order. Now we don t need to worry about the order. Example: A bag contains 5 blue, 4 black, 6 yellow, and 5 white marbles. If two blues and a black marble were drawn, how many ways can they be arranged? 5
Permutations (Repetitions) Example: You selected a full house with 3 tens and 2 aces. In how many ways could the cards have been drawn? License Plates Example: For many years, the State of California used 3 letters followed by 3 digits on its automobile license plates. (a) How many different license plates are possible with this arrangement? (b) When the state ran out of new numbers, the order was reversed to 3 digits followed by 3 letters. How many new license plate numbers were then possible? (c) Several years ago. the numbers described in b were also used up. The stale then issued plates with I letter followed by 3 digits and then 3 letters. How many new license plate numbers will this provide? Combination 2 4 495 6
Combination versus Permutation A permutation is when order matters. Lining people up. Selecting people from a group and assigning them positions. A combination is when order does not matter. Selecting cards Picking people for a team. vs. Permutations Example: For each problem, tell whether permutations or combinations should be used to solve the problem. (a) How many 4-digit code numbers arc possible if no digits are repeated? (b) A sample of 3 light bulbs is randomly selected from a batch of 5. How many different samples are possible? (c) In a baseball conference with 8 teams, how many games must be played so that each team plays every other team exactly once? (d) In how many ways can 4 patients be assigned to 6 different hospital rooms so that each patient has a private room? Locker Combination If you have a locker combination does the order matter? Example: There are 5 people in a class. How many ways can 4 of them be selected to work on a project together? 7
Example: There are 52 cards in a standard deck of cards. In how many ways can you select 5 cards? Example: There are 20 people in Club West. Five students must be selected to work on an activity. (a) How many ways can this be done? (b) How many ways can this be done if Joe must be in the group? (c) What if the question was change to at most 3 people need to selected for the group? Example: In how many ways can you select a full house with tens over aces? Example: When selecting a five card hand: (a) How many possible hands are there with two hearts? (b) How many possible hands are there with all face cards? (c) How many possible hands are there where all 5 cards are of the same suite? 8
Example: How many possible full houses are there? Example: The senate contains 5 democrats, 47 republicans, and independent. How many ways can a committee be formed if it must contain 3 democrats and 2 republicans? Recall that probability can be calculated as = Applications of Counting Principles = () () For more complicated situations we can use the multiplication principle, combinations, and permutations to calculate probability. 9
Example: There are 52 cards in a standard deck of cards. What is the probability of getting four of a kind? : Example : What is the probability of being dealt a full house? : Using the Multiplication Principle: Using the Multiplication Principle: Example : There are 20 people in Club West, 2 of which are women. Five students must be selected to work on an activity. If the people were randomly chosen, (a) What is the probability of there being 3 women and 2 men? Example: The senate contains 5 democrats, 47 republicans, and independent. A committee of 0 must be formed. If the people were randomly chosen, (a) What is the probability of there being 3 democrats, 6 republicans, and independent? (b) What is the probability of there being all women? (b) What is the probability of there being 9 democrats, and republican? 0
Example: Find the probability of each. (a) A 5-card hand with two hearts? We don t need no stinkn order! Remember when we did probability before and I kept saying, in that order. (b) A 5-card hand with all face cards? (c) How many possible hands are there where all 5 cards are of the same suite? We don t need it any more. Example: A bag of marbles contains 3 red, 5 blue, 7 yellow, and 5 green marbles. If 3 are drawn, what is the probability of (a) Drawing 3 marbles and they are yellow, green, and blue? Example: The Environmental Protection Agency is considering inspecting 6 plants for environmental compliance: 3 in Chicago, 2 in Los Angeles, and in New York. Due to a lack of inspectors, they decide to inspect 2 plants selected at random, this month and next month, with each plant equally likely to be selected, but no plant is selected twice. What is the probability that Chicago plant and Los Angeles plant are selected? (b) Drawing or more green marbles?
Example: From a group of 50 accountants at a firm 5 are selected to write a review of management policies. (a) How many possible groups of 5 can be made? Example: During the 988 college football season, the Big Eight conference ended the season with a perfect progression. (a) How many games did the 8 teams play? (b) Assuming no ties, how many different outcomes are there for all the games together? (b) What is the probability that Jessica Smith, one of the accountants, is selected? (c) In how many ways could the eight teams end in a perfect progression? (d) Assuming each team had an equal chance of winning, what is the probability of a perfect progression. Definition Binomial Flipping a coin. Do a behavior, don t do a behavior. Faulty, not faulty. 2
MAT 4 - Chapter 8 Binomial Formula Example: You flip a coin 0 times. What is the probability of getting 4 heads? You flip a coin 3 times. What is the probability that you get two heads?, Here are the possible outcomes. The probability could be thought of as 2 2 2 2 2 2 2 2 2 3 2 3 2, 2 3 2 2 Example: Over a long period of time it has been observed that a given rifleman can hit a target on a single trial with probability equal to.8. Suppose that he has four shots at a target: Example: A bag contains 4 red, 5 green, 7 blue, and 4 white marbles. (a) What is the probability of drawing 0 marbles and three of them are blue? (a) What is the probability that he will hit it twice? (b) What is the probability that he will hit it at least twice? (b) What is the probability of drawing 0 marbles and at least four of them are red? 3
Example: A company produces light bulbs. It has found that out of every 0,000 is faulty. If a sample of 30 light bulbs is taken what it the probability that at least one is faulty? Distributions; Expected Value Making a Distribution A probability distribution represents the probabilities of all possible simple events. To make a probability distribution, do the following: Step : List all possible simple outcomes. A Distribution Example 8.5.: All possible outcomes and a probability distribution for the sum when two dice are rolled are shown below. Possible outcomes Step 2: Step 3: Identify outcomes that represent the same event and determine the probability of each event. Make a table listing each event and probability. 4
A Distribution Example: Create a probability distribution for the experiment of flipping 3 coins. Random Variable Event 3 Heads H, 2 T 2H, T 3 Tails Outcomes* *Technically a probability distribution would not include an outcomes row. We include it here to organize our work. Any experiment where we can assign numerical values. Rather than probability of 3 heads in 5 tosses we analyze the number of heads in 5 tosses. This could take on the values of 0,, 2, 3, 4, 5 The number of heads is a random variable. Looking at all the values of a random variable we can calculate the probability of each value and create a probability distribution. A Distribution Example 8.5.3: Create a probability distribution for the random variable of the number of heads in 5 tosses. # Heads 0 Heads Heads 2 Heads 3 Heads 4 Heads 5 Heads 0.0325 0.5625 0.325 0.325 0.5625 0.0325 To calculate these use the binomial probability. 0 Head: 5 0 (.5) (.5) = 0.0325 4 Head: 5 4 (.5) (.5) = 0.5625 Distribution (Histogram) Distribution for Number 5 Tosses 0.35 0.3 0.25 0.2 0.5 0. Head: 5 (.5) (.5) = 0.5625 2 Head: 5 2 (.5) (.5) = 0.325 3 Head: 5 3 (.5) (.5) = 0.325 5 Head: 5 5 (.5) (.5) = 0.0325 0.05 0 0 2 3 4 5 Number of Heads 5
MAT 4 - Chapter 8 Distribution (Histogram) Here is the distribution and histogram if we had 2 tosses. Expected Value Consider two or more events, each with its own value and probability. = ( + + 2 ) ( 2 ) The expected value can be thought of as the theoretical result if you averaged a HUGE number of trials of an event. Expected Value Book formula. Expected Value, Insurance Example: Suppose an automobile insurance company sells an insurance policy with an annual premium of $200. Based on data from past claims, the company has calculated the following probabilities: An average of in 50 policyholders will file a claim of $2,000. An average of in 20 policyholders will file a claim of $,000. An average of in 0 policyholders will file a claim of $500. Assuming that the policyholder could file any of the claims above, what is the expected value to the company for each policy sold? = $200 $2000 $000 $500 = $60 50 20 0 *It is $200 to buy a policy and you must buy one, hence a probability of, if you are going to make a claim. Also, as this is money coming into the company it is positive. The other amounts are claims, which is money going out and therefore negatives. The result tells the company that they expect to make $60, on average, for every $200 policy that they sell. Slide 7-64 6
Expected Value, Insurance What if the company in the last example sold 200,000 such policies? Expected Value, Coin Example: Let s play a game. We flip a coin. If it is heads I pay you $2 and if it is tails you pay me $3. What is the expected value (from your perspective)? Expected Value, Deal or No Deal? Example: You are playing Deal or No Deal?. The following cases are left. The banker offers you $278,000. Should you take it? Expected Value (Raffle) Example: A group is raising money by selling raffle tickets for $5 each. The prizes are $2500 vacation package $500 local wine collection 4 $00 gift cards to a restaurant It is known that a 2,000 tickets were sold. Create a probability distribution. Prize $2495 $495 $95 -$5 /2000 /2000 4/2000 994/2000 7
Raffle Expected Value What is the expected value of one ticket? Example: Through the season we see that Sue is a 65% free-throw shooter. What is the expected value if she shot 8 free-throws in last nights game? 8