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WARM-UP

WARM-UP. Given that P(A)=.35, P(B)=.85, P(AUB)=.8 Find P(AB) P(A|B) P(B|A) 2. Determine if the events A and B are independent. Counting and Permutations. GROUP ACTIVITY.

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WARM-UP

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  1. WARM-UP • Given that P(A)=.35, P(B)=.85, P(AUB)=.8 • Find P(AB) • P(A|B) • P(B|A) 2. Determine if the events A and B are independent

  2. Counting and Permutations

  3. GROUP ACTIVITY Break into groups and solve the word problem provided to each group. Each word problem is different, so you should only be talking to the people in your group

  4. FUNDAMENTAL COUNTING PRINCIPAL • Given a combined action, or event in which the first action can be performed in n₁ ways, the second action can be performed in n₂ ways, and so on, the total number of ways in which the combined action can be performed is the product n₁∙n₂…..

  5. INDIVIDUAL ACTIVITY • You are setting up a special playlist in your iPod. You trying to put 5 songs in an order to your liking. List the songs you are putting in the playlist, and say how many ways you can have the songs listed in that playlist

  6. PERMUTATION • A permutation of a set of n objects is an ordered arrangement of all “n” objects

  7. TOTAL NUMBER OF PERMUTATIONS OF “N” OBJECTS • The total number of permutations of “n” objects, denoted P(n,n) is given by P(n,n)=n∙(n-1)∙(n-2)∙…1

  8. FACTORIAL NOTATION • For any natural number “n”, n!=n∙(n-1)∙(n-2)∙….1 also n!=n∙(n-1)! • For the number 0, 0!=1 • So, n!=P(n,n)

  9. QUESTION: • How many 5-digit numbers can be formed using the digits 2,4,6,8, and 9 without repetition? With repetition?

  10. WITHOUT REPETITION I can only choose from 5 numbers. Every time I choose a number, I will have one less number to choose from the next time 5∙4∙3∙2∙1=20∙6=120

  11. WITH REPLACEMENT I have 5 numbers to choose from. Every time I choose a number, I am allowed to choose that number again. So, I will ALWAYS have 5 numbers to choose from 5∙5 ∙5 ∙5 ∙5=5⁵

  12. GROUP WORK • There are 7 people and 3 chairs. How many different ways can I put 7 people in 3 distinct chairs? • I have 3 balls and 2 cups. How many ways can I put 3 balls in 2 distinct cups? (only 1 ball can go in each cup). • http://www.khanacademy.org/math/probability/v/permutations

  13. Number o f permutations of a set of “n” objects taken k at a time The number of permutations of a set of “n” objects taken k at a time, denoted P(n,k) given by P(n,k)=n∙(n-1)∙(n-2)…[n-(k-1)] =n!/(n-k)!

  14. SOLVE • P(6,6) • P(10,7) • 5! • P(4,3) • (8-5)! • 9!/4!

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