Communicating Quantitative Information

1
Communicating Quantitative Information

• Numbers, lottery, Powerball
• Probability and odds
• Homework Postings!

2
Variance

• Standard deviation is square root of variance
• Consider a set of numbers x1, x2,
• The mean is (x1x2.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.

3
Chance newsletter

• Was a consortium of colleges
• Newsletter
• Wiki jointly edited newsletter
• http//chance.dartmouth.edu/chancewiki/index.php/M
  ain_Page

4
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 whendata is grouped

5
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

6
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

7
Initial Data
AA 100
AA 50
RKO 500
Sony 1000
Sony 250
8
Sort

• Under Data, click on Sort and then Column B and
  then descending

Sony 1000
RKO 500
Sony 250
AA 100
AA 50
9
How to do totals by company?
Sony 1000
RKO 500
Sony 250
AA 100
AA 50


Sony
RKO
AA

• Add in the 3 distinct companies
• sumif(range, criteria, sum_range)
• SUMIF(A1A5,A8,B1B5)
• The 's means this will copy correctly to the
  other rows.

10
Sony 1000
RKO 500
Sony 250
AA 100
AA 50


Sony 1250
RKO 500
AA 150
11
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

12
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) ?
• ????

13
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)

14
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!

15
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

16
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

17
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.

18
Wrong way

• could not do the problem by
• P(A or B the first drawing) 2/5
• P(A or B the second drawing) ????

19
Make a tree

• A tree is a diagram used to organize/analyze/prese
  nt situations
• A B C D
  E
• BCDE ACDE ABDE ABCE ABCD

20 outcomes 2 successes
20
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)

21
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 / (XY)
• If probability is p, odds for are p versus
  (1-p).Odds against are (1-p) versus p.

22
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)

23
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!

24
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.

25
Chance project archives

• www.dartmouth.edu/chance
• put in lottery numbers mob or
• http//www.dartmouth.edu/chance/chance_news/recen
  t_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

26
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

27
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
• 26 254 26100 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.

28
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.

29
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

30
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.

31
Probability

• The number of different red ball possibilities is
  42
• Total number of different outcomes
  iscombinations(49,5) 42
• ((4948474645) / (54321)) 42
• 80089128

32
Expectation

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

33
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

34
Trick question

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

35
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