Chance News 47
It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge.
Pierre Simon Laplace
Théorie Analytique des Probabilités, 1812,
Their visual acuity is only fractionally – not geometrically – better than that of the common primates from which they were engineered.
When do you draw the line? When do you take action to avoid that logarithmic point where things take off exponentially?
Submitted by Margaret Cibes
Andrew Gelman has an interesting article regarding the statistics of elections. He starts with
“Presidential elections have been closer in the past few decades than they were for most of American history. Here's a list of all the U.S. presidential elections that were decided by less than 1% of the vote:
Funny, huh? Other close ones were 1844 (decided by 1.5% of the vote), 1876 (3%), 1916 (3%), 1976 (2%), 2004 (2.5%).
Four straight close elections in the 1870s-80s, five close elections since 1960, and almost none at any other time.”
Perhaps more interesting is his take on House and Senate elections:
From the graph, “the rate of close elections in the House has declined steadily over the century. If you count closeness in terms of absolute votes rather than percentages, then close elections become even rarer, due to the increasing population. In the first few decades of the twentieth century, there were typically over thirty House seats each election year that were decided by less than 1000 votes; in recent decades it's only been about five in each election year.”
Another way of putting it: “Consider that, in the past decade, there were 2,175 elections to the United States House of Representatives held on Election Days 2000, 2002, 2004, 2006 and 2008. Among these, there were 41 instances — about 1.9 percent — in which the Democratic and Republican candidates each received 49 percent to 51 percent of the vote (our calculations exclude votes cast for minor parties). In the 1990s, by contrast, there were 65 such close elections. And their number increases the further one goes back in time: 88 examples in the 1950s, 108 in the 1930s, 129 in the 1910s.”
1. A contention mentioned in the NYT article for this bifurcation of opinion is: “as the economy has become more virtual, individuals can now choose where to live on an ideological rather than an occupational basis: a liberal computer programmer in Texas can settle in blue Austin, and a conservative one in the ruby-red suburbs of Houston.” Argue for and against this assertion of coupling ideology and occupational mobility.
2. Gelman and Silver end their NYT article with “Elections like those in New York’s 20th district or in Minnesota, as contentious as they are, actually hark back to a less divisive era in American politics.” Explain the seeming paradox of a close race indicating less divisiveness.
3. As of this posting, the Minnesota senate seat is unfilled despite a manual recount, a canvassing board followed by a so-called election contest and awaits the results of the appeal to the Minnesota Supreme Court—possibly further. See When Does ‘Close Become Too-Close-to-Call? for an analysis of error rates and how likely it is that the real winner would be Norm Coleman instead of Al Franken who currently leads by 312 votes out of about 2.9 million cast. The Minnesota Supreme Court will hear oral arguments on June 1, 2009.
Submitted by Paul Alper
Bayes theorem in the news
In a letter to the editor of the Miller-McCune magazine, headed "For those of You Who Paid Attention in Statistics Class" , Professor Howard Wainer, of The Wharton School, writes about the "mathematical reality ... that so long as only a very small minority of people commit crimes and the criminal justice system is fair ... there will always be a very large proportion of innocent people convicted." He provides a hypothetical example, with calculations based on Bayes theorem.
Some interesting challenges to his assumptions, both from the editor and online bloggers, are included . Professor Wainer's response to an editor's comment includes the following medical example.
Each year in the U.S., 186,000 women are diagnosed, correctly, with breast cancer. Mammograms identify breast cancers correctly 85 percent of the time. But 33.5 million women each year have a mammogram and when there is no cancer it only identifies such with 90 percent accuracy. Thus if you have a mammogram and it results in a positive (you have cancer) result, the probability that you have cancer is: 186,000/(186,000+3.35 million) = 4%.
So if you have a mammogram and it says you are cancer free, believe it. If it says you have cancer, don't believe it.
The only way to fix this [is to] reduce the denominator. Women less than 50 (probably less than 60) without family history of cancer should not have mammograms.
1. Consider Professor Wainer's calculation of the probability (4%) of having cancer given a positive mammogram result. Do you agree with the numbers used?
2. Consider Professor Wainer's advice to a person who receives a diagnosis of being cancer free. What is the probability of being cancer free if one receives a negative mammogram result? Do you agree with his advice?
3. Comment on Professor Wainer's advice for "women less than 50."
Submitted by Margaret Cibes