Probability 4-6 Counting Fundamental Counting Rule Permutations Combinations
Fundamental Counting Rule (Space Rule) For a sequence of two or more events m and n The first event occurs m ways and the second occurs n ways, the events can occur m x n ways
Example 1 Assume that a criminal found your credit card and claims that all the digits were randomly generated. What is the probability of getting your social security number? Is the criminal s claim that your number was randomly generated likely to be true?
Solution to Ex. 1 9 digits in SSN. Each of the 9 spaces can be filled with 10 possible numbers. * * * * * * * * 1,000,000,000 possibilities 1/1,000,000,000
Example 2 In computer science, a byte is defined to be a sequence of 8 bits. Each bit must be a 0 or 1 (this is why it s called a binary code, there are only two choices). How many different bytes are possible?
solution 2^8=256
Example 3 We have already talked about how the order that questions are asked can influence how survey subjects answer questions. To avoid this, pollsters often rearrange the order questions are presented. If a polling agency plans to conduct a consumer survey by asking 5 questions, how many different versions of the survey could be constructed?
Factorial Rule The factorial symbol (!) denotes the product of decreasing whole numbers. 4! = 4 3 2 1 = 24 Factorial Rule a collection of n different items can be arranged in order n! different ways.
Factorial Rule and Routing Problems Routing problems often involve the factorial rule.verizon want to route calls through the shortest networks. Federal Express wants to find the shortest routes between 3 cities. How many different routes are possible?
Routes to National Parks During the summer, you want to visit six national parks: Glacier, Yellowstone, Yosemite, Arches, Zion, and Grand Canyon. You want to plan the most efficient route and list all of them. How many different routes are possible?
Permutations Rule The number of permutations (or sequences) of r items from n available items (not allowing repetition) is n! n P r ( n r)!
Conditions for Permutations There must be a total of n different items available. (The permutations rule above does not apply if some of the items are identical) Select r of the n items (without replacement) Rearrangements of the same items are considered different sequences. Order is taken into account.
Exacta Bet An exacta bet in horse is when you select the first and second place finishers in order. The Kentucky Derby had 20 horses. If I randomly select two horses for an exacta, what is the probability that I win?
Solution N=20 horses, we select 2(r) without replacement. n! 20! 380 ( n r)! (20 2)! 1/380
Permutations Rule (when some items are identical to others) If there are n items with n alike, n alike,... n alike, 1 2 k the number of permutations of all n items is n! n! n!... n! 1 2 k
Assigning seats There are students in your class. of them are males and are females. How many ways can you be arranged?
Combinations Rule The number of combinations of r items from n different items is n! n C r ( n r)! r!
Conditions for Combinations There must be n different items available. Select r of the n items (without repetition). Rearrangements of the same items are considered the same groupings. Order is not taken into account (if the same items are present in different order, the groupings are considered the same.)
Example of Permutations and Combinations The Board of Trustees at a college has 9 members. Each year, they elect a 3-person committee to oversee buildings and grounds. Each year, they also elect a chairperson, vice chairperson, and secretary.
Choosing the Type of Problem a. When the board elects the buildings and grounds committee, how many different 3-person committees are possible? b. When the board elects the 3 officers, how many different slates of candidates are possible?
Summary Is there a sequence of events in which the first can occur m ways, the second n ways, etc...? If so, use the fundamental counting rule. M*n*... Are there n different items and are all of them to be used in different arrangements? If so use factorial rule, n!
Are there n different items and are only some of them to be used in different arrangements? If so use Permutations. npr Are there n items with some of them identical to eachother and is there a need to find the total number of different arrangements of all those n items? If so use repetition rule. n!/n1!n2!... Are there n different items, with some of them selected and is there a need to find the total number of combinations, where order is irrelevant? If so, use Combinations. ncr
HMWK 1-21 odd