Here is a real-world problem, dealing with a very timely topic
(primary elections and poll numbers) that
illustrates conditional probabilities, Bayes' Rule, and the Total
Probability Rule. The problem is based on the following
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USA Today article*, published on 2/15/2000,
during the height of the 2000 Primary Election season.
(Amazingly enough, even though the article is 8 years old, it is still
available online, at the above link!)

Bush leads McCain 49%-42% among likely voters in South Carolina, according to a USA TODAY/CNN/Gallup Poll Friday through Sunday. The survey suggests Bush has slowed McCain's surge, particularly among Republicans.The article describes a poll, taken among South Carolina voters shortly before the Republican primary in that state. The polling data cited in the article shows the following:Bush shifted gears after his loss in New Hampshire two weeks ago, playing up his conservative credentials and attacking McCain's record in the Senate. Analysts say the tactics helped Bush regain strength in South Carolina, where he once held a huge lead. "Traditional Republicans seem to be swinging more strongly to Bush now that McCain is no longer as new and fresh as he was two weeks ago," Furman University political scientist Jim Guth said.

McCain needs South Carolina to prove that New Hampshire was not a fluke and build momentum for the next primary, Feb. 22 in Michigan. Bush needs to prove he is the solid front-runner.

The Republicans debate tonight at 9 p.m. ET (live, CNN). The poll shows that McCain's best hope remains among independents and Democrats, who can vote in the primary:

Among Republicans, Bush leads McCain 59%-34%.

Independents support McCain 54%-36%. Democrats who say they intend to cross party lines also support McCain, 61%-27%.

The poll shows 60% of voters will be Republicans, 34% independents and 6% Democrats. "This is a guessing game until we see who actually turned out," said Brad Gomez of the University of South Carolina. The poll of 552 likely voters has a margin of error of +/-5 percentage points.

- Bush leads McCain 49 % - 42 % among likely voters
- Among Republicans, Bush leads McCain 59 % - 34 %
- Independents support McCain 54 % - 36 %
- Democrats who say they intend to cross party lines (and vote in the Republican primary) also support McCain, 61 % - 27 %
- 60 % of the voters are Republicans, 34 % Independents, and 6 % Democrats.

From the given data and the context of the problem, one can assume the following:

- A voter is either a Democrat, a Republican, or an Independent.
- Since the voting percentages for Bush and McCain don't add up to 100 %, one can assume that the remaining votes went to other candidates or were left blank. It makes sense to lump all of these votes together into a single group, "Other", that represents the complement of the Bush and McCain votes.

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Questions
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- The first statistic given (Item 1 above: Bush leads McCain 49 % - 42 % among likely voters) turns out to be redundant. Derive that statistic from the other data given (items 2 - 5).
- Among McCain supporters, what percentage are Independents?
- Among Bush supporters, what percentage are Independents?

To solve these questions, proceed as follows:

- Identify the relevant events and introduce appropriate notation for these events.
- Express the data given in items 2 - 5 above in terms of this notation.
- Similarly, express the probabilities sought in the above questions in terms of this notation.
- Now use appropriate probability rules (e.g., Bayes' Rule, Total Probability Rule) to derive the probabilities sought from those that were given.

Back to the Math 408 Course Webpage

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Last modified: Wed 30 Jan 2008 08:16:56 AM CST
A.J. Hildebrand
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