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Communicating Quantitative Information Numbers, lottery, Powerball Probability and odds Homework: Postings!
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.
Chance newsletter • Was a consortium of colleges • Newsletter • Wiki: jointly edited newsletter http://chance.dartmouth.edu/chancewiki/index.php/Main_Page
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
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
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
Sort • Under Data, click on Sort and then Column B and then descending
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.
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
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) = ? • ????
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)
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!
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
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
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.
Wrong way • could not do the problem by • P(A or B the first drawing) = 2/5 • P(A or B the second drawing) = ????
Make a tree • A tree is a diagram used to organize/analyze/present situations AB C D E BCDE ACDE ABDE ABCE ABCD 20 outcomes: 2 successes
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)
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.
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)
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!
Numbers • The probability of getting any particular number is 1/1000 • Number determined using total bet at certain horse race (or races) • 3rd, 5th, 7th 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.
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 Mob State runners 25 stores 5 controller 5 expenses 15 bank 10 tax relief 30 pay out 60 pay out 50
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…
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.
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.
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
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.
Probability • The number of different red ball possibilities is 42 • Total number of different outcomes iscombinations(49,5) * 42 ((49*48*47*46*45) / (5*4*3*2*1)) * 42 80089128
Expectation • jackpot * (1/80089128) plus all (prize * probability) of lesser levels • What am I leaving out????????
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
Trick question • I have two children.One is a boy.What is the probability that I have two boys?
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