Transcript Communicating Quantitative Information

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.ph
p/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
Initial Data
AA
100
AA
50
RKO
500
Sony
1000
Sony
250
Sort
• Under Data, click on
Sort and then Column
B and then
descending
Sony
1000
RKO
500
Sony
250
AA
100
AA
50
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.
Sony
RKO
1000
500
Sony
AA
AA
250
100
50
Sony
RKO
AA
Sony
1000
RKO
500
Sony
250
AA
100
AA
50
Sony
1250
RKO
500
AA
150
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
A
B
BCDE ACDE
C
ABDE
D
E
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
runners
controller
bank
pay out
State
25
5
10
60
stores
expenses
tax relief
pay out
5
15
30
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 is
combinations(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