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Chapter 3 Probability

Chapter 3 Probability . Initial Objective: Develop procedures to determine the # of elements in a set without resorting to listing them first. 3-6 Counting . Order is important . Worksheet: Coloring outfits to determine the total number of combinations possible.

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Chapter 3 Probability

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  1. Chapter 3 Probability Initial Objective: Develop procedures to determine the # of elements in a set without resorting to listing them first. 3-6 Counting

  2. Order is important • Worksheet: Coloring outfits to determine the total number of combinations possible. • http://illuminations.nctm.org/lessons/combinations/Combinations-AS-ShortsandShirts.pdf

  3. Possible Results • Yellow shirt, Brown shortsYellow shirt, Black shortsYellow shirt, Green shortsYellow shirt, Purple shortsOrange shirt, Brown shortsOrange shirt, Black shortsOrange shirt, Green shortsOrange shirt, Purple shortsBlue shirt, Brown shortsBlue shirt, Black shortsBlue shirt, Green shortsBlue shirt, Purple shortsRed shirt, Brown shortsRed shirt, Black shortsRed shirt, Green shortsRed shirt, Purple shorts

  4. Website Bobby Bear -Students can pick an outfit for Bobbie Bear and customize the outfit similar to the one on the activity sheet. • http://illuminations.nctm.org/ActivityDetail.aspx?ID=3

  5. Discussion Questions • How did your prediction compare to your actual answer? How do you explain this? • Which method would be more efficient for finding the total number of outfits: multiplying, drawing a tree diagram, or making a table? • Which method would be more useful for identifying the different combinations (outfits) possible: multiplying, drawing a tree diagram, or making a table?

  6. Answer Example • In designing a computer, if a byte is defined to be a sequence of 8 bits, and each bit must be a 0 or 1, how many different bytes are possible? • Answer: Since each bit can occur in 2 ways (0 or 1) and we have a sequence of 8 bits, the total # of different possibilities is given by 2x2x2x2x2x2x2x2= 256

  7. Factorial Rule-n different items can be arranged in order n! different ways Ex. 5! = 5x4x3x2x1= 120 0! =1 • Key is on your calculator • Note: The factorial rule reflects that fact that the 1st item maybe selected n different ways; the 2nd item maybe selected n-1 ways, and so on.

  8. Factorial Examples • Example: How many possible ways routes are there to 3 different cities? • Example: What about possible routes each of the 50 states?

  9. What if you don’t what to include all of the items available?Order is still important!

  10. Note: rearrangements of the same items to be different. Look at page 155 for an example.

  11. What if we tend to select r items from n available items, but do not take order into account? • We are really concerned with possible combinations.

  12. B: Order is not important • When different ordering of the same items are to be counted separately, we have a permutation problem, • but when different orderings are not to be counted separately, we have a combination problem.

  13. Run through page 156 to 158 Could you image not having these counting techniques, it would take hours and hours to come up with all the possibilities.

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