Chapter 5 Possibilities and Probability Counting Permutations Combinations Probability

5.1 Counting Example 5.1: Ice cream cones Flavor: chocolate, vanilla, strawberry Cones: sugar, regular How many different varieties?

Tree diagram sugar choc reg sugar van reg sugar str reg 3*2=6 choices

Rule 1: Multiplication Rule A choice consists of 2 distinct steps 1 st step can be made in m different ways For each of these, 2 nd step can be made in n different ways Then the whole choice can be made in m*n ways

Example horses in a race How many ways can we pick a first and second place finisher? Answer: 4  3=12 You can get it from a tree diagram as well

Choice of 1 st placechoices of 2 nd place 4*3=12 ways A B C D A B C A B D A C D B C D

Rule 2: Generalized Multiplication Rule A choice consists of k steps; Step 1 can be made in n 1 ways; Step 2 can be made in n 2 ways; … … Step k can be made in n k ways ; Then whole choice can be made in n 1 n 2 …n k ways

Example horses in a race How many ways can we pick a 1 st, 2 nd, and 3 rd place finisher?

Example 5.4 A multiple choice exam has 5 questions Each question has 4 possible answers What is the number of ways to answer the exam?

Example 5.5 How many license plates can be formed with 3 letters followed by 3 numbers?

Example 5.6 How many ways can we write the letters O W L ?

5.2 Permutations The number of ways to order r of n objects n P r =n(n-1)(n-2)…(n-r+1) (application of multiplication rule)

Example 5.7 How many ways can we put 3 cards from a deck of cards in order? (52)(51)(50)=132,600

Notation: n!=(1)(2) … (n-1)(n) The number of ways to order (permute) n of n objects is n P n = n(n-1)(n-2)…(1) = n! n! is called n-factorial 3!=(1)(2)(3)=6 4!=(1)(2)(3)(4)=3!(4)=24 5!=4!(5)=120 (0!=1)

Example 5.8 How many ways can we order 5 horses? 5!=120

Example 5.9 How many ways can we arrange the letters OLWS?

The Number of Permutations The Number of Permutations

Exercise How many different ways can we inject 3 of 15 mice with 3 different doses of a serum?

Exercise 5.21 In optics kits there are 5 concave lenses, 5 convex lenses, 2 prisms, and 3 mirrors. how many different ways can a person choose 1 of each kind?

Exercise 5.27 In how many different ways can a television director schedule a sponsor’s six different commercials during a telecast?

5.3 Combinations How many ways can we choose 3 of 5 candidates to be in the final election? Here the order of choosing the 3 finalists doesn’t matter!

Think this way … … When we did worry about the order, there are (5)(4)(3)=60 ways ABC ACB BAC BCA CAB CBA ABD ADB BAD BDA DAB DBA BCD BDC... … …

and Each choice of 3 candidates has 3!=6 ordered versions ABC  ABC ACB BAC BCA CAB CBA

Therefore ABC ACB BAC BCA CAB CBA  ABC ABD ADB BAD BDA DAB DBA  ABD BCD BDC...  BCD … … The options are decreased by a factor of 6 compared to the ordered options

The number of ways to pick 3 out of 5

The Number of Combinations The number of unordered ways to choose r of n objects is (n choose r)

Pascal ’ s Triangle

Example 5.10 Choose 5 cards from a deck

Example flavors of ice cream. Choose 2. Order doesn’t matter.

Example candidates in a primary election. Choose 2 for a final election

Exercise Calculate the number of ways in which a chain of ice cream stores can choose 2 of 12 locations for new franchises.

Exercise A computer store carries 15 kinds of monitors. Calculate the number of ways in which we can purchase 3 different ones.

Exercise In planning a garden we have 5 kinds of bushes to choose from and 10 kinds of flowers. How many ways can we choose 2 kinds of bushes and 4 kinds of flowers?

5.4 Probability In a deck of cards what is the probability of picking an ace? What is meant by “Probability”? Frequency interpretation: The probability of an event happening is the proportion of times that event would occur in the long run.

Facts Each card is equally likely to be chosen for a shuffled deck of cards If there are “n” equally likely possibilities and “s” of these are a “success”, then the probability of a success is s/n P(ace)= 4/52=1/13 P(red)=26/52=1/2

Example 5.13 Draw 2 cards from a deck. What is the probability that we get 2 aces?

Ideas Each card is equally likely to be selected if one card is selected All pairs of cards are equally likely to be selected if only two cards are selected Any three cards are equally likely to be selected if only three cards are selected … … (we call these randomness)

Solution to Example 5.13 # of possible ways to pick 2 cards # of ways to pick 2 aces Probability P(2 aces)

Example 5.14 Pick up 3 cards, what is the probability of getting 2 aces and 1 king? Let’s work it out!

Step 1 Step 2 Step 3 Probability P(2 aces and 1 king)=s/n= 24/22100 # of ways to pick 2 aces out of 4 # of ways to pick 1 king out of 4

Example 5.15 Pick up 5 cards. What is the probability of getting 3 aces?

Step 1 Step 2 Step 3 Probability P(3 aces in 5 cards )=s/n= # of ways to pick 3 aces out of 4 # of ways to pick other 2 cards

Example 5.16 Roll a red die and a white die. Find the probability that sum=3. red die: 1, 2, 3, 4, 5, 6 white die: 1, 2, 3, 4, 5, 6 n= # of outcomes = (ways for red to land)*(ways for white to land) = 6×6=36 pairs s= # of ways sum=3 (R=1, W=2), (R=2, W=1) =2 Probability=2/36=1/18