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Transcript of Probability
total number n P r restriction total number n! (n-r)! n! n C i r c u l a r P r e m u t a t i o n Reflective Permutation Independent Events: choices DO NOT affect the outcome of other situations example: Akiko has six dresses, fives pairs of shoes, and two coats. How many combinations of these items are possible? Dependent Events: choices DO affect the outcomes of other outcomes example: There are three people making a speech. How many orders can they go in? Person 1 Person 2 Person 3 Once a person goes, then the remaining two go. 3!= 3 x 2 x 1= 6 6 dresses x 5 pairs of shoes x 2 coats= 60 combinations example: If four coins are flipped, how many total outcomes are possible? H T H T H T H T 2 x 2 x 2 x 2 =16 HHHH
THTT 16 Outcomes What is the probability of getting exactly all H? HHHH
THTT 1 16 What is the probability of at least getting one tail? HHHH TTTT HHTH THTH
HHHT TTHT TTHT HTHT
HHTT THHT HTHT THHT
HTTT THHH THTT HTTH 15
16 What is the probability of getting at most one tail? HHHH TTTT THTH HHTH
HHHT TTTH HTHT TTHT
HHTT TTHH THHT HTHH
HTTT THHH HTTH THHT 5
16 n! n / 2 n C r n! (n-r)! r! n C r x (p) x (q) r n-r Binomial Probability Permutations with Restrictions P (n,r) = n!
(n-r)! Repeated Permutations n! p! x q! Basic Counting Principle n x n x n x n. . . 2 2 2 2 a. The first letter has to be C.
Find the number of ways that you can arrange the word CREST using the following restrictions:
C R E S T 4 x 3 x 2 x 1= 24 b. The first letter cannot be vowel. C R E S T 4 x 4 x 3 x 2 x 1 =96 Permutations with Restrictions
P (n,r) = n!
(n-r)! Find the number of ways that you can arrange the word: MATHEMATICS 11!
2! X 2! X 2!= 4,989,600 P (n,n) = n! Permutation C (n,r) = n!
(n-r)! r! Combination Order doesn't matter.
3 x 2 x 1= 6 From a pool of 10 candidates, the offices of president, vice president, secretary, and treasurer will be filled. In how many different ways can the office be filled? The state of North Carolina makes license plates with 3 letters and three digits.
a. No Restrictions
b. The letters and numbers must vary
Mini Quiz From a list of 12 books, how many combinations of 5 books can be selected? How many ways can 10 people sit around a campfire? How many ways can 8 keys be arranged on a key chain?