# Probability Material

## David C. Howell

In this section I will try to present material that helps to highlight some of the important concepts of probability theory. It is difficult to come up with good material that will interest most students, so bear with me.

• The three door problem is an interesting problem to play with, and it illustrates several important things, though I am putting it in here more because it was fun to write up than because it fits well with either text.
• The following is taken directly from Chance News (1996--5.04), quoting, I believe, from the Daily Telegraph (2/4/96) "Hawkings writes in 'Radio times' that he thinks that gambling profits are a pretty sleazy way to raise money even for good causes. The real interest in this article is the comment that 'Statisticians have determined that if you buy a national Lottery ticket on a Monday, you are 2500 times more likely to die before the Saturday draw than land the jackpot."
• Think about that for a moment.
• How do you suppose "Statisticians" ever came to that conclusion?
• What important variables would have to be taken into account before making such a pronouncement?
• I share Hawking's view of lotteries, but not his appeal to nameless "statisticians."
• You often wake up to a prediction from the weather reporter that "the chances of rain today are 40%"
• What does that statement mean to you?
• What do you think that the weather reporter meant by that statement?
• Would you expect that 40% of the towns in your listening area would have rain?
• What kind of probability is this--subjective, relative frequency, or analytic?
• Would you be more likely to take your umbrella if she said that the chances of rain were 80%?
• In a few chapters we are going to make statements of the form "If the hypothesis that it doesn't make any difference whether you stay or switch (in the three-door problem) were true, the probability that we would win 11 times out of the 15 trials on which we switched is .0417."
• What does it mean when we say that any probability is .0417?
• What would we mean to we say that the probability of as least 11 wins is .0512?
• If we actually won 11 times out of 15 switched trials, could we conclude that the probability of the hypothesis that switching doesn't make any difference is .0417?
• If we actually won 11 times out of 15 switched trials, could we conclude that the probability of the hypothesis that switching doesn't make any difference is .0512?
• The answer to the two previous questions is "No," but lots of people make that mistake. Why is it a mistake, nonetheless? (This example might not be a good one because we can calculate what the probability of winning when we switch actually is. Make up a similar example where the truth of the situation is a little vaguer. Does that help answer the last question?)
• If you're interested in an interesting class discussion of basic probability, you might try the tire problem athat appears as item #20 on the 4/1/96 Chance News. The multiple solutions (given alternative assumptions) are interesting.
• An interesting paper by Fell (1995) on the relationship between drugs, alcohol, and automobile accidents offers a lot of opportunity for probability examples. That paper is available at http://raru.adelaide.edu.au/T95/paper/s14p1.html, and also at http://www.uvm.edu/~dhowell/StatPages/More_Stuff/s14p1.html.

Send mail to: David.Howell@uvm.edu)

Last revised: 7/11/98