Download "Week 6" notes with annotations made in lecture on Wednesday

STA 220H1F LEC0201 Week 6: Probability and Relative Frequency Probability in the Headlines: From http://www.thestar.com/news/crime/article/1044561‐‐
police‐mum‐about‐falling‐homicide‐rates Toronto police don t want to talk about this, even though the numbers
reflect positively on them. Homicides are down, and down substantially.
This no longer seems a city living in fear as it was in 2005 when gang
warfare spilled out into the streets in the “summer of the gun.” The city s
32nd homicide occurred on Tuesday, compared to 38 at this time last year.
With only four months to go in 2011, Toronto is on pace to record its lowest
homicide total since 1999 when there were 47 slayings. After rising
through the early part of the decade and peaking at 86 murders in 2007,
rates have fallen for three straight years and may drop again by year end.
There were 70 homicides in 2008, 63 in 2009 and 61 in 2010.
1 From http://www.huffingtonpost.co.uk/alex‐morrison/the‐
euromillions‐winner‐c_b_961514.html The Euromillions Winner Could Be You - But It
Probably Won't Be
Posted: 16/9/11 08:44 GMT
"IT COULD be you," the old National Lottery advert proclaimed, a huge disembodied hand
hovering over your house in a manner which made poverty seem like the least of your
worries.
You can't deny the lottery's logic - you could win. But you could also fall off the Empire State
Building and land safely in a lorry full of your favourite flavour of jam, driven by someone
who turns out to be the love of your life.
Your chance of winning the Euromillions jackpot with a single ticket is one in 116,531,800.
Such large numbers are difficult to comprehend, so here's some context.
• You'd have a better chance of calling a random phone number in Mexico (assuming
each of the country's 112 million residents, regardless of age, had one phone) and
getting the exact person you wanted. Buying another ticket increases your chances, but
only by the same degree as giving the Mexican phone thing another go.
• You have a greater chance of dying in the minutes between buying your ticket and the
draw being made than you do of winning.
• If you're aged 15-24, you have to buy your ticket 12 minutes or less before the draw to
have more chance of winning than of dying in the meantime (based on 2008 death rates
and population estimates for England and Wales from the Office for National Statistics).
• With 13 minutes to go, the odds of expiring overtake those of scooping the jackpot though those are based on the entire population and you can improve your chances with
a healthy diet, regular exercise and by avoiding dangerous pursuits like table tennis and
lawn bowls.
• Over 85s have to buy a ticket one second before the draw to have a better chance of
winning than dying before it starts - two seconds before and the Grim Reaper is once
again favourite, and his odds rapidly improve in the short time the draw itself takes.
• Unfortunately, Euromillions ticket sales stop about 90 minutes before the draw, so
you've always got more chance of kicking the bucket than winning the jackpot at the
point of purchase.
• British people have an estimated one in 685,000 chance of drowning in a bath and a
one in 10,000,000 chance of being killed by lightning.
Helpfully, www.euro-millions.com points out those odds are based on a lifetime, not any
single day. So, the website says: "When you compare Euromillions odds with these specific
odds for various grisly deaths, you can see that you're actually far more likely to win a
lottery jackpot this Tuesday or Friday than you are to meet your maker in any (one)
spectacular fashion."
So don't be put off.
If the metaphorical (rather than the actual) lightning strikes and you win the Euromillions,
your life is nowhere near long enough for the probabilities to balance out.
You'd have to buy a lottery ticket twice a week for more than a million years for that - so, if it
is you, just don't re-invest all your winnings on lottery tickets.
2 From http://www.cbc.ca/news/canada/prince‐edward‐
island/story/2011/09/03/nb‐alc‐not‐concerned.html?cmp=rss Atlantic Lottery Corp. not concerned about
insider wins
CBC News
Posted: Sep 3, 2011 2:38 PM AT
Lottery jackpots won by retailers and other insiders in Atlantic Canada are
frequent but not a concern, the Atlantic Lottery Corporation says. This week,
CBC News reported that although jackpot wins by insiders remain high in
Atlantic Canada — the Atlantic Lottery Corporation said it's not concerned.
"Reporting those numbers are the outcome of a process that was put in place
for player protection and that process works," said Paula Dyke, director of
public affairs for ALC.
Winnings connected to the corporation nearly tripled last year in Atlantic
Canada. In 2010-11, the non-arm’s length winners for big prizes jumped to
11.9 per cent compared to 4.3 per cent the previous year. That means that
almost one in eight prizes worth more than $25,000 went to someone
connected to ALC or its suppliers. "Non-arm's length winners" is a term that
refers to people who work for ALC or its suppliers, or people who own or work
at stores that sell tickets or family members of the same group.
Maureen Tingley, a mathematician at the University of New Brunswick, called
the jump “wacky” and said more explanation is needed.
From the Globe and Mail September 11, 2004 3 Random (to a statistician) Individual outcomes are uncertain, but there is a structure to how often outcomes occur in a very large number of repetitions. Using Probability to Quantify the Uncertainty of a Random Circumstance A random experiment is a natural phenomenon, or scientific experiment, or sampling experiment, whose outcome cannot be predicted. Each occurrence of a random experiment is a trial. Probability is a number between 0 and 1 assigned to each of the outcomes of a random experiment. Outcomes with probability 1 are sure to occur and outcomes of probability 0 are sure not to occur. A sample space, , for a random experiment is the set of all possible outcomes of the experiment. Example: Toss a coin twice 4 An event is a collection of one or more outcomes. An event occurs if any outcome making up that event occurs. Example: Toss a coin twice Event : At least one head The complement of an event , denoted , occurs if does not occur. It contains all outcomes not in . Example: Toss a coin twice Event : At least one head Events and sample spaces are often illustrated by Venn Diagrams. 5 Combining events: •
contains all outcomes in or (or both). •
contains all outcomes which are in both and . Two events that have no outcomes in common are said to be mutually exclusive. Mutually exclusive events cannot occur at the same time. Mutually exclusive events are sometimes called disjoint. Example: Toss a coin twice Event : Heads on the first toss Event : Tails on the first toss 6 Interpretations of Probability Interpretation 1: Long Run Relative Frequency If you toss a coin, what’s the probability that it will turn up heads? In the 18th century, Compte de Buffon tossed a coin 4040 times and got 2048 heads. 2048
Relative frequency of heads
0.507 4040
In 1900, Karl Pearson threw 12,012 heads in 24,000 tosses. 12012
Relative frequency of heads
0.5005 24000
While imprisoned during World War II, mathematician John Kerrich kept track of his results when tossing a coin 10,000 times. His results: 7 This is the empirical or experimental approach to probability. In situations that we can imagine repeating many times, we can define the probability of an outcome as the long run relative frequency that the outcome occurs. Each individual outcome is unpredictable, but the long run average is predictable. Law of Large Numbers As the number of repetitions of a random experiment increases, the chance that the relative frequency of an event will differ from the true probability of the event by more than any small number approaches 0. But beware of the Gambler’s Fallacy. The Law of Large Numbers does not hold in the short run. Even though you got 3 heads in 3 tosses of coin, you are not more likely to get tails next time. (Is tossing a coin really random? Watch Program 15 of Against All Odds at http://www.learner.org/resources/series65.html to see probabilist (and former magician) Persi Diaconis flip a coin and get heads every time.) 8 The long run relative frequency conception can be used to give an estimate of the actual (theoretical) probability. Interpretation 2: Theoretical Probability Theoretical probability is based on a model for how a random phenomenon occurs. The simplest model is based on equally likely outcomes. If a sample space has possible outcomes, and all outcomes in a sample space are equally likely, then the probability of any one outcome is 1 . Then Calculations based on theoretical probability models and equally likely outcomes are used to calculate the probabilities of the outcomes of lotteries and games involving dice and cards. 9 Interpretation 3: Subjective Probability A true story … In 1977 a PanAm jumbo jet and a KLM jumbo jet collided on an airport runway in the Canary Islands, killing 581 people. A wire service report on the incident included a statement from the chairman of the American National Transportation Safety Board saying that statistics showed that the chances of two jumbo jets colliding on the ground were about 6 million to one. A statistician, Terry Speed, wrote to the chairman to ask how that figure had been calculated. The response: “…the chairman’s statement concerning the chances of two jumbo jets colliding has no statistical validity nor was it intended to be a rigorous or precise probability statement. The statement was made to emphasize the intuitive feeling that such an occurrence indeed has a very remote but not impossible chance of happening.” This is an example of subjective or personal probability, the degree of belief an individual has about an event. (Sometimes it’s more valid than others.) 10 The probability of an event, , is denoted . Rules for Valid Probabilities 1. The probability of any event is between 0 and 1. 0
1 2. The probability of all of the outcomes in the sample space is 1. 1 Two events are independent of each other if knowing that one will occur (or has occurred) does not change the probability that the other occurs. (Otherwise, the events are dependent.) Suppose and are dependent events. | , read “the probability of given ”, is the conditional probability that the event occurs, given that we know has occurred or will occur. If and are independent, then |
. Example: Toss a coin twice Event = 2 heads Event = First toss is a head versus Event = head on the second toss Event = First toss is a head 11 If and are disjoint events, must they be independent? Rules for Finding Probabilities 3.
1
4. Addition rule: (a)
(b) For mutually exclusive events: Example: Toss a coin twice Event = 2 heads Event = 2 tails = coin comes up the same both times 12 5. Multiplication rule: (a)
| (b) If and are independent events Example: Toss a coin twice Event = first toss is heads Event = second toss is heads Another Example: Draw 2 cards from a deck without replacement. What is the probability you get 2 red cards? 13 Calculating conditional probabilities |
Example: Suppose there has been a crime and it is known that the criminal is a person within a population of 6,000,000. Further, suppose it is known that in this population only about one person in a million matches the DNA found at the crime scene. So there are 6 people in the population who have this DNA type. Someone in custody has this DNA type. What is the probability he is actually innocent? Answer: Using conditional probability ideas (to get the same answer), let Event = DNA of randomly chosen person matches DNA at the crime scene Event = person randomly selected is innocent of the crime |
5⁄6,000,000
6⁄6,000,000
5
6
A different conditional probability: What is the probability of getting a DNA match for an innocent person? |
5/6,000,000
5,999,999/6,000,000
.0000083 14 Another Example: Using Tables The table below shows some of the results of an early (1998) study on internet communication, in particular, “flaming” in chat rooms. The investigators were interested in whether being criticized would make one more likely to criticize others. Has Been Personally Criticized
Has Criticized 19 Others Has Not 23 Criticized Others Total 42
Has Not Been Personally Criticized
8 Total
27 143 166 151
193
Assume that this table is representative of all chat room users. Suppose that chat room user is randomly selected. Define the events: event that the individual has criticized others event that the individual has been personally criticized by others Estimate and interpret: •
15 •
•
•
|
•
|
16