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Communicating Quantitative Information Numbers, lottery, Powerball Probability and odds Homework: Postings!

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Variance Standard deviation is square root of variance Consider a set of numbers: x1, x2, … The mean is (x1+x2+….xn) / n. Let m be the mean. The variance is ((x1-m) 2 + (x2-m) 2 + …) –sum of the squares of the difference of each value from the mean. –Taking the square means the values on each side of the mean contribute to the total. –The is a measure of the spread of the numbers.

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Chance newsletter Was a consortium of colleges Newsletter Wiki: jointly edited newsletter http://chance.dartmouth.edu/chancewiki/index.ph p/Main_Page

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Spread sheet assignment Consider centering headings for columns Consider inserting gridlines Consider color, bold, other markup, especially for headings of columns or rows Do not display more significant digits than warranted –limit digits to the right of the decimal point –be consistent with money (2 digits or none) Consider what is the appropriate form of graph: –pie charts for parts of a whole –line charts for something in which horizontal axis is a scale, not individual items –stacked or clustered bars when…data is grouped

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Spreadsheet, continued Make your name be in print! Make title be in print! Proof read!!! –Catch mistakes like treating heading line as data –Catch typos Invest in staple or paperclip! Move charts around to get all on one page

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Spreadsheets, cont. Use Excel to sort (up or down) –Data / Sort Sort rows based on values in one or more columns Use built-in functions –For example, conditional sum is sumif

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Initial Data AA100 AA50 RKO500 Sony1000 Sony250

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Sort Under Data, click on Sort and then Column B and then descending Sony1000 RKO500 Sony250 AA100 AA50

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How to do totals by company? Add in the 3 distinct companies sumif(range, criteria, sum_range) =SUMIF(A$1:A$5,A8,B $1:B$5) The $'s means this will copy correctly to the other rows. Sony1000 RKO500 Sony250 AA100 AA50 Sony RKO AA

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Sony1000 RKO500 Sony250 AA100 AA50 Sony1250 RKO500 AA150

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Probability P(event_E) is chances of event_E happening divided by total number of possible outcomes. Coin throw (assuming 'fair' coin): P(heads) = ½ Dice (die) throw (assuming 6 sided, fair die) P(3) = 1/6

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More probabilities Die throw P(1 or 3 or 5) = 3/6 P(3 or 4) = ? Probability: put names of everyone in this room in a hat and draw out –P(my_name) = ? –P(student) = ? –????

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Probabilities Independent events: like the coin toss, no dependence one on the other Probability of two independent events are the product of the two probabilities Coin toss: –probability of head followed by a head are (1/2)*(1/2)

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Probabilities for combined events Throw a coin two times: P(Head Head) = ¼ P(Head Tail) = ¼ P(Tail Head) = ¼ P(Tail Tail) = ¼ Note: Can derive the probability another way: 4 outcomes, each equally likely, so P of each is 1/4 Outcomes may not be equally likely, but the probabilities of all outcomes always total 1!

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Different problem Throwing a coin twice, what is the probability that you get 1 head and 1 tail, you do not care about the order: P(Head Head) = ¼ P(Head Tail) = ¼ P(Tail Head) = ¼ P(Tail Tail) = ¼ P(1 each) = ¼ + ¼ = ½

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Hat Hat contains A, B, C, D, E Probability of drawing A and B, don't care about the order are –P(A and then B) = 1/5 times ¼ = 1/20 =.05 –P(B and then A) = 1/5 times ¼ = 1/20 =.05 –P(either of these two events) =.05 +.05 =.1

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Two events Probability of two independent events both happening are product of probabilities. Probability of either of two events happening is the sum of probabilities. NOTE: the probability of drawing out of a hat A first and then B AND drawing out B first and then A is zero!!!!! Both these events cannot happen.

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Wrong way could not do the problem by –P(A or B the first drawing) = 2/5 –P(A or B the second drawing) = ????

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Make a tree A tree is a diagram used to organize/analyze/present situations A B C D E BCDE ACDE ABDEABCE ABCD 20 outcomes: 2 successes

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Probability versus odds P(event) = event/(all outcomes) P(success) = (successful outcomes)/(all outcomes) odds to succeed = (successful outcomes) / (failure outcomes) odds to fail (odds against) = (failure outcomes) / (successful outcomes)

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Odds Even odds = 1 to 1. There are even odds to throw a head with a fair (unbiased) coin Odds against throwing a 1 using regular dice is 5 to 1 Odds against throwing 1 or 2 is 4 to 2 If odds against outcome are given X to Y then probability of outcome is Y / (X+Y) If probability is p, odds for are p versus (1-p). Odds against are (1-p) versus p.

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Numbers Assuming each digit (0, 1, 2,…9) are equally likely, the probability of any particular 3 number pattern is 1 / (10 * 10 * 10) Think of how many different numbers there are (writing 0 as 000, 1 as 001, 12 as 012, and so on)

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Expectation …. of a bet is (value of winning) * (probability of winning) A bet is fair if the stake = expectation Bet $1 to get (payoff) $2 if you toss heads 2 * (1/2) = 1 This is a fair bet!

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Numbers The probability of getting any particular number is 1/1000 Number determined using total bet at certain horse race (or races) –3 rd, 5 th, 7 th bet called the 3-5-7 –numbers, policy, bolita… The payoff (in the old days, by the mob) was (typically) 600 to 1. This was NOT a fair bet. –popular numbers sometimes had lower payoffs but….the payoff for state lotteries are typically 500 to 1 for similar situations.

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Chance project archives www.dartmouth.edu/~chance –put in lottery numbers mob or http://www.dartmouth.edu/~chance/chance_news/recent_news/chance _news_10.08.html MobState runners 25stores 5 controller 5expenses15 bank10tax relief30 pay out60pay out50

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Permutations & Combinations To draw 3 specific letters from 26 tiles holding letters of the alphabet, in specific order is 26 * 25 * 24 ABC is not the same as ACB This is called a permutation. If order does not matter, this is called a combination. To calculate how many combinations, determine how many ways you can shuffle 3 distinct things and divide by that number…

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How many…. permutations are there of 3 things? 3 * 2* 1 = 6 SO…. for drawing tiles holding letters, A to Z (no repeats), drawing 3 tiles, combinations are (26 * 25 * 24) / 6 26 * 25*4 = 26*100 = 2600 Why divide? Think of grouping all the outcomes with the same tiles. Each group has 6 elements. How many groups? The total divided by 6.

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Recall old example Draw from 5 letters: how many different combinations: (5 * 4) / 2 2 ways to shuffle = re-order 2 things 10 different combinations, each equally likely, so probability of any one (say the A and B combination) is 1/10.

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Powerball 5 white balls from 49 distinct balls 1 red ball from 42 distinct balls Jackpot: match 5 white balls (any order) plus red ball prizes for (lower) levels of matching –5 white balls and not the red ball –4 white balls and the red ball –4 white balls and not the red ball, …. –0 white balls and the red ball

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Powerball history If no one wins, the money goes to the next competition. Organization increased the number of balls to decrease the odds to produce more times when jackpots built-up. Bigger jackpots drew more ticket sales.

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Probability The number of different red ball possibilities is 42 Total number of different outcomes is combinations(49,5) * 42 ((49*48*47*46*45) / (5*4*3*2*1)) * 42 80089128

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Expectation jackpot * (1/80089128) plus all (prize * probability) of lesser levels What am I leaving out????????

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Expectation You may have to share the jackpot –probabilities go up as number of tickets go up Jackpot is less than they advertise –immediate cash (less) versus annuity (at full amount spread over 25 years) will talk about time value of money later must pay taxes See Chance, same issue as lottery/numbers

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Trick question I have two children. One is a boy. What is the probability that I have two boys?

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Homework Postings Responses to postings Make sure you understand –percentage –issue of definition, concept of model –mean, median, mode, standard deviation (variance), range –probability, odds, expectation, payoff

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