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PROBABILITY

Learn how to apply the Fundamental Counting Principle, Permutations, and Combinations to solve various probability problems effectively. Discover real-life applications of counting principles for better understanding.

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PROBABILITY

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  1. Section 3.4: The Counting Principle PROBABILITY

  2. Objective: To be able to understand and apply the counting principle. The Fundamental Counting Principle: If event 1 occurs M ways and event 2 occurs N ways, then events 1 and 2 occur in succession M*N ways. (This principle can be extended to more than 2 events.) Ex. 1 Social Security Numbers • How many exist? • How many SS numbers start with 3 even numbers?

  3. c. How many SS numbers have no repeats? Ex. 2 Pin numbers for ATMs • How many exist? • How many begin and end with an odd number? • How many start with the number 4?

  4. Permutations: an ordered arrangement of objects. (ORDER MATTERS!) The number of different permutations of n distinct objects is n! (0! = 1) Ex. 3 How many different possible standings are there in the PAC10 assuming that there are no ties? Ex. 4 How many different ways can you arrange 7 different baseball cards on a table from left to right?

  5. Permutations of n objects taken r at a time: where n > r Ex. 5 A club has 8 members and selects a president and vice president. How many ways can this happen? Ex. 6 A horse race has 7 horses in it. How many different trifecta bets exist? Trifecta is picking 1st , 2nd and 3rd places in that order.

  6. Distinguishable Permutations: The numbers of permutations of n objects where are one type and are a second type and so on, Ex. 7 There are 7 empty lots in a neighborhood. The builder wants to build 3 Manor homes, 2 Colonial homes and 2 Ranch homes. How many ways can these homes be arranged in the 7 lots?

  7. Combinations of n objects taken r at a time: A combination of r objects taken from a group of n objects without regard to order is denoted where n > r ORDER DOES NOT MATTER Ex. 8 How many ways can I select 5 students to go to the office from our class of 15? Ex. 9 How many ways can you pick 3 songs to play from a your music library of 7 songs?

  8. APPLICATIONS OF COUNTING PRINCIPLES: Ex. 10 Using a standard deck of cards, find the probability of being dealt 5 diamonds. Ex. 11 Find the probability of being dealt a 3 of a kind.

  9. Ex. 12 Find the probability of winning a trifecta bet when there are 6 horses in a horse race. Ex. 13. Repeat exercise 12 with the bet boxed.

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