Probability Alexander Riccio Jose Deliz But first, a quick review Permutations Combinations If there are A ways for one event to occur, and B ways for a second event to occur, there are A • B ways for both to occur. There are six ways to roll a die and two ways to flip a coin. There are twelve ways to roll a die and THEN flip a coin. = 12 X A club of ten people are choosing a governing committee.

Three people are going to be on this committee.

The order in which they are selected does not matter. Counting principle A selection in which order does not matter

Represented as (nCr), where N is the number of choices possible, and R is the size of the selection nCr A selection in which order DOES matter The permutations of the letters A and B A B + AB

BA AB and BA are different.

There are two Permutations. Combinations and Permutations example problems How many different licence plates of 3 symbols (letters and digits) can be made using 2 letters for each? The notation for Permutation:

nPr

n is the total number of choices

r is the number of objects chosen The forumula for Permutation: n! __________ (n-r)! We are required to choose 3 objects

Two of which have 26 possible choices

One of which has 10 possible choices ___X___X___ 26 26 10 ___X___X___ (26 26 10) x 3 But also, we have to multiply by 3, because the number can be in any one of 3 digits # Letter Letter 26 26 10 26 10 26 10 26 26 = 20,280 Licence plates From a set of 9 different books, 4 are to be selected and arranged on a shelf. How many arrangements are possible? 9 4 P = the number of arrangements

= the number of arrangements

= the number of arrangements

= the number of arrangements 9!

(9-4)! ____ 9!

5! __ 3024 Find the number of ways the letters in the word "hubbub" can be arranged Special rule applies: like terms P = --------------- n!

n1!,n2!,.... P = --------------- 6!

2!3! P = 60 Hubbub because.... 2 Us & 3 Bs X = 8 Sample Space (#1) (H,1)(H,2)(H,3)(H,4) (T,1)(T,2)(T,3)(T,4) (#3) Combinations The forumula for Combination: r!(n-r)! n! The notation for combination

n r

n is the total number of choices

r is the number of objects chosen __________ (#5) (#8) A sample space is all the possible outcomes of an experiment

An Event is the specific outcomes of an experiment Experimental Theoretical Experimental probability changes as you test out each experiment. The more trials you run, the closer the experimental probability is to the theoretical probability. Theoretical probability is the mathematical liklehood of something occuring Jack is playing a game in which each player must roll a die. To win the game Jack must roll a number greater than 4. What is the probability that Jack will win on his next turn? There are only 2 ways for Jack to win the game.

Therefore the probability of Jack winning is: 2 _ 6 A group of students were asked to name their favorite menu item at the school cafeteria.

Of those polled, 12 chose hot dogs, 27 chose subs, and 11 chose pizza. If we asked all 100 students in the grade, how many could be expected to choose pizza as their favorite item? (hot dog) + (subs) + (pizza) = 12 27 11 50 P 11/50 .22 22% = = = Pizza # of students who favor pizza = (.22)(100) = 22 The End C

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# Probability

By Alexander Riccio, and Jose Deliz

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