Copyright © 2011 Pearson Education, Inc. Probability: Living with the Odds.

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Copyright © 2011 Pearson Education, Inc. Probability: Living with the Odds

Copyright © 2011 Pearson Education, Inc. Slide 7-3 Unit 7E Counting and Probability

7-E Copyright © 2011 Pearson Education, Inc. Slide 7-4 If we make r selections from a group of n choices, a total of different arrangements are possible. Example: How many 7-number license plates are possible? Arrangements with Repetition There are 10 million different possible license plates.

7-E Copyright © 2011 Pearson Education, Inc. Slide 7-5 Permutations We are dealing with permutations whenever all selections come from a single group of items, no item may be selected more than once, and the order of arrangement matters. e.g., ABCD is different from DCBA The total number of permutations possible with a group of n items is n!, where

7-E Copyright © 2011 Pearson Education, Inc. Slide 7-6 If we make r selections from a group of n choices, the number of permutations (arrangements in which order matters) is The Permutations Formula Example: On a team of 10 swimmers, how many possible 4-person relay teams are there? There are possible relay teams!

7-E Copyright © 2011 Pearson Education, Inc. Slide 7-7 The Permutations Formula Example: If an international track event has 8 athletes participating and three medals (gold, silver and bronze) are to be awarded, how many different orderings of the top three athletes are possible? There are 336 different orderings of the top three athletes!

7-E Copyright © 2011 Pearson Education, Inc. Slide 7-8 Combinations Combinations occur whenever all selections come from a single group of items, no item may be selected more than once, and the order of arrangement does not matter e.g., ABCD is considered the same as DCBA If we make r selections from a group of n items, the number of possible combinations is

7-E Copyright © 2011 Pearson Education, Inc. Slide 7-9 Example: If a committee of 3 people are needed out of 8 possible candidates and there is not any distinction between committee members, how many possible committees would there be? The Combinations Formula There are 56 possible committees!

7-E Copyright © 2011 Pearson Education, Inc. Slide 7-10 Probability and Coincidence Example: What is the probability that at least two people in a class of 25 have the same birthday? The answer has the form Although a particular outcome may be highly unlikely, some similar outcome may be extremely likely or even certain to occur. Coincidences are bound to happen.

7-E Copyright © 2011 Pearson Education, Inc. Slide 7-11 Birthday Coincidence The probability that all 25 students have different birthdays is

7-E Copyright © 2011 Pearson Education, Inc. Slide 7-12 The probability that at least two people in a class of 25 have the same birthday is Birthday Coincidence P(at least one pair of shared birthdays) = 1 – P(no shared birthdays) ≈ 1 – ≈ ≈ 57% The probability that at least two people in a class of 25 have the same birthday is approximately 57%!

7-E Copyright © 2011 Pearson Education, Inc. Slide 7-13 Birthday Coincidence 1 0 y = Probabilities x = People in Room What are the probabilities that someone in a room of x people will have a birthday in common with someone else in that room?

7-E Copyright © 2011 Pearson Education, Inc. Slide 7-14 Birthday Coincidence 1 0 y = Probabilities x = People in Room What are the probabilities that someone in a room of x people will have a birthday in common with someone else in that room?

7-E Copyright © 2011 Pearson Education, Inc. Slide 7-15 Birthday Coincidence 1 0 y = Probabilities x = People in Room What are the probabilities that someone in a room of x people will have a birthday in common with someone else in that room?

7-E Copyright © 2011 Pearson Education, Inc. Slide 7-16 Birthday Coincidence 1 0 y = Probabilities x = People in Room What are the probabilities that someone in a room of x people will have a birthday in common with someone else in that room?

7-E Copyright © 2011 Pearson Education, Inc. Slide 7-17 Birthday Coincidence 1 0 y = Probabilities x = People in Room What are the probabilities that someone in a room of x people will have a birthday in common with someone else in that room?

7-E Copyright © 2011 Pearson Education, Inc. Slide 7-18 Birthday Coincidence 1 0 y = Probabilities x = People in Room How many people in the room would be required for 100% certainty? What are the probabilities that someone in a room of x people will have a birthday in common with someone else in that room?