Slide 1 Math 1520, Lecture 23. This lecture covers mean, median, mode, odds, and expected value.

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1 Slide 1 Math 1520, Lecture 23 This lecture covers mean, median, mode, odds, and expected value.

2 Slide 2 Mean, Median and Mode Mean, Median and mode are 3 concepts used to get a sense of the central tendencies of a group of numerical data. Take a list of numbers, and arrange them in increasing order: x 1, x 2,..., x n 1. The Mean is the sum of the numbers divided by how many there are. 2. The Median is The middle number if there is an odd number of entries or The mean of the two middle numbers if there is an even number of entries. 3. The mode is the number (or numbers) in the group that occur(s) most frequently, if any. (There may be no mode, a unique mode or more than one mode.) 1, 4, 5, 7 has no mode 1, 3, 3, 3, 5 has the mode of 3 2, 2, 2, 5, 5, 8, 8, 8 has modes 2 and 8

3 Slide 3 iclicker Question The time in minutes that it takes a person to get to work each morning over a 4 day period is 10, 11, 9, 14 minutes. What is the mean? A B. 11 C. 12 D. 12.5

4 Answer to Question The time in minutes that it takes a person to get to work each morning over a 4 day period is 10, 11, 9, 14 minutes. What is the mean? A B. 11 is the correct answer. C. 12 D. 12.5

5 Slide 4 iclicker Question The time in minutes that it takes a person to get to work each morning over a 4 day period is 10, 11, 9, 14 minutes. What is the median? A B. 10 C. 11 D. 12.5

6 Answer to Question The time in minutes that it takes a person to get to work each morning over a 4 day period is 10, 11, 9, 14 minutes. What is the median? A is the correct answer. B. 10 C. 11 D. 12.5

7 Slide 5 iclicker Question The time in minutes that it takes a person to get to work each morning over a 5 day period is 8, 10, 11, 9, 14 minutes. What is the median? A B. 10 C. 11 D. 9.5

8 Answer to Question The time in minutes that it takes a person to get to work each morning over a 5 day period is 8, 10, 11, 9, 14 minutes. What is the median? A B. 10 is the correct answer. C. 11 D. 9.5

9 Slide 6 Odds In gambling and common speech, people frequently refer to the idea of odds: The odds the Yankees will win the World Series this season are 2 to 5. The actual definition of what this means is as follows Definition If P (E) is the probability of an event E occurring then The odds in favor of E occurring are P (E) 1 P (E) The odds against E occurring are 1 P (E) P (E) (P (E) 1) (P (E) 0) If the odds in favor of E are expressed as a fraction a/b then we say that: the odds in favor of E are a to b We can also reverse this to convert odds to probability using the following result: Theorem If the odds of an event occurring are a to b then the probability of E occurring is P (E) = a a + b 1. Find the odds for drawing a club from a standard deck. 2. If The odds the Yankees will win the World Series this season are 2 to 5, what is the probability that they will win? 3. Prove the theorem.

10 Slide 7 iclicker Question What are the odds against against drawing a two or a three from a standard deck? A. 2/13 B. 13/2 C. 11/2 D. 2/11

11 Answer to Question What are the odds against against drawing a two or a three from a standard deck? A. 2/13 B. 13/2 C. 11/2 is the correct answer. D. 2/11

12 Slide 8 Expected Value, Definition Description In general, the expected value is a number associated with a random variable X. If trials of an experiment are done repeatedly, the average values of the observed values of X will get closer to the expected value as the number of trials get larger and larger. Formal Definition Let X denote a random variable that assume the values x 1, x 2,..., x n with associated probabilities p 1, p 2,..., p n. Then the expected value of X is denoted by E(X) and is given by E(X) = x 1 p 1 + x 2 p x n p n Example Suppose that the number of cars waiting in line at a toll booth is counted at each 5 minute interval and the following data is obtained Number of Cars Frequency of Occurrence Turn this into a probability distribution. 2. A person is approaching the toll booth. How many cars can they expect to find waiting in line?

13 Slide 9 iclicker Question A Roulette wheel has 38 slots with 18 slots colored red, 18 slots colored black and 2 slots colored green. A one dollar bet is placed on red. If the ball lands on red, you get your dollar back and win one more dollar, if the ball does not land on red, you lose the dollar. What is the expected value of net winnings on a $1 bet on red? A. $2/38 B. $18/38 C. $20/38 D. $2/38

14 Answer to Question A Roulette wheel has 38 slots with 18 slots colored red, 18 slots colored black and 2 slots colored green. A one dollar bet is placed on red. If the ball lands on red, you get your dollar back and win one more dollar, if the ball does not land on red, you lose the dollar. What is the expected value of net winnings on a $1 bet on red? A. $2/38 is the correct answer. B. $18/38 C. $20/38 D. $2/38

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