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Counting Techniques (Dr. Monticino)

Counting Techniques (Dr. Monticino). Overview. Why counting? Counting techniques Multiplication principle Permutation Combination Examples Probability examples. Why Counting?.

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Counting Techniques (Dr. Monticino)

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  1. Counting Techniques(Dr. Monticino)

  2. Overview • Why counting? • Counting techniques • Multiplication principle • Permutation • Combination • Examples • Probability examples

  3. Why Counting? • Recall that if each outcome of an experiment is assumed to be equally likely, then the probability of an event is k/n • where k is the number of elements in the event and n is the number of elements in the sample space • So to calculate the probability of an event, we need to be able to count the number of elements in the event and in the sample space

  4. Multiplication Principle • Multiplication principle. Suppose that an experiment can be regarded as a series of k sub-experiments. Such that the first sub-experiment has n1 possible outcomes, the second sub-experiment has n2 possible outcomes, and so on. Then the total number of outcomes in the main experiment is • n1x n2 x ... x nk • Examples • Flip a coin and roll a die • Roll 5 die; or roll a single die five times

  5. Permutation • Factorial. n! (read “n factorial”) equals • Permutation. The number of ways to select r objects, in order, out of n objects equals

  6. Examples • How many ways are there to do the following • Line up 10 people • Select a President, VP and Treasurer from a group of 10 people • Sit 5 men and 5 women in a row, alternating gender

  7. Combination • Combination. The number of ways to select r objects out of n objects when order is not relevant equals

  8. Examples • How many ways are there to do the following • Select 3 people from a group of 10 • Select 7 people from a group of 10 • Get exactly 5 heads out of 12 coin flips

  9. Probability Examples • Select three people at random from a group of 5 women and and 5 men • What is the probability that all those selected are men? • What is the probability that at least one women is chosen? • What is the probability that at least two women are chosen?

  10. Probability Examples • Flip a fair coin 3 times • What is the probability that 3 heads come up? • What is the probability that at least 1 tail occurs? • What is the probability that exactly 2 tails occur? • What is the probability that at least 2 tails occur?

  11. Probability Examples • Play roulette 3 times • What is the probability that red comes up every time? • What is the probability that black comes up at least once? • What is the probability that black comes up exactly two times? • What is the probability that black comes up at least two times?

  12. Probability Examples • Flip a fair coin 10 times • What is the probability that 10 heads come up? • What is the probability that at least 1 tail occurs? • What is the probability that exactly 8 tails occur? • What is the probability that at least 8 tails occur?

  13. Probability Examples • Play roulette 20 times • What is the probability that red comes up every time? • What is the probability that black comes up at least once? • What is the probability that black comes up exactly 18 times? • What is the probability that black comes up at least 18 times?

  14. Probability Examples • Roll a fair die 5 times • What is the probability that an ace comes up all five times? • What is the probability that an ace occurs at least once? • What is the probability that an ace occurs exactly 3 times? • What is the probability that an ace occurs at least 3 times?

  15. Probability Examples • To win the jackpot in Lotto Texas you need to match all six of the numbers drawn (5 numbers are selected from numbers 1 to 44 and the sixth is selected separately from 1 to 44) • What is the probability of winning if you buy one ticket? • What is the probability of winning if you buy five tickets? • Is it better to buy five tickets in one Lotto drawing or a single ticket in five successive Lotto games?

  16. Assignment Sheet • Read Chapter 15 carefully • Redo all problems from lecture • Not to turn in… (Dr. Monticino)

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