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Understanding Counting Methods for Computing Probabilities in Modern Science

Explore the significance of counting techniques in calculating probabilities, including multiplication rule, permutations, and combinations. Learn how to apply these methods in real-life scenarios and grasp their importance in modern science.

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Understanding Counting Methods for Computing Probabilities in Modern Science

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  1. First lecture fsalamri Fatenalamri

  2. Old saying • “probability is the most important concept in modern science especially as nobody has slightest notation what it means” • Counting techniques • 1) multiplication rule 2) computation 3)permutation fsalamri

  3. P(A) = N(A)\N(S) fsalamri • Note: these are called “counting methods” because we have to countthe number of ways A can occur and the number of total possible outcomes.

  4. Example 1: You draw one card from a deck of cards. What’s the probability that you draw an ace? fsalamri

  5. Toss 1: 2 outcomes Toss 2: 2 outcomes 22 total possible outcomes: {HH, HT, TH, TT} H H T H T T • Example • What's the probability of 2 heads when tossing a coin? fsalamri

  6. Example • What’s the probability of rolling a die then tossing a coin? fsalamri

  7. Combinations— Order doesn’t matter Permutations—order matters! With replacement Without replacement Without replacement Counting methods for computing probabilities fsalamri

  8. Formally, “order matters” and “with replacement” use powers • Example • How many license plates can we have using three letters followed by three digits ? • Alphabet 26 digit # 10 • = • =(17576)(1000) • =17,576,000 fsalamri

  9. Permutation: The number of ways in which a subset of objects can be selected from a given set of objects, where order is important • Example 1.6 pg5 • Example • From a club of 24 members, a President, Vice President, Secretary, Treasurer and Historian are to be elected. In how many ways can the offices be filled? fsalamri

  10. Answer Example 1.4. How many permutations are there of all three of letters a, b, and c? Answer: = n!\(n − r)! =3!\0! = 6 fsalamri

  11. Combinitions • Combination: The number of ways in which a subset of objects can be selected from a given set of objects, where order is not important. • A Combination is an arrangement of items in which order does not matter. fsalamri Since the order does not matter in combinations, there are fewer combinations than permutations.  The combinations are a "subset" of the permutations.

  12. To find the number of Combinations of n items chosen r at a time, you can use the formula c is denoted by fsalamri

  13. Example 1.7 pg 5 • Example • To play a particular card game, each player is dealt five cards from a standard deck of 52 cards. How many different hands are possible? • Answer fsalamri

  14. Guidelines on Which Method to Use fsalamri

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