This document discusses a case study involving Jane Blaylock, a professional golfer who was disqualified from a tournament. She sued for damages using a statistical method called expected value to estimate how much money she could have earned. Her lawyers used data from her past performances to calculate probabilities of various scores she may have achieved. This statistical evidence convinced the jury to award her damages. The document then provides an activity on estimating probabilities in the game of craps using dice rolls. [/SUMMARY]
11. Example 7.1 Getting good grades
Finding discrete probabilities
North Carolina State University posts the grade distributions for its courses online. Students in
Statistics 101 in the fall 2003 semester received 21% A's, 43% B's, 30% C's, 5% D's, and 1% F's.
Choose a Statistics 101 student at random. To “choose at random” means to give every student the
same chance to be chosen. The student's grade on a fourpoint scale (with A = 4) is a random
variable X.
The value of X changes when we repeatedly choose students at random, but it is always one of 0, 1,
2, 3, or 4. Here is the distribution of X:
The probability that the student got a B or better is the sum of the probabilities of an A and a B. In
the language of random variables,
P(X ≥ 3) = P(X = 3) + P(X = 4)
= 0.43 + 0.21
= 0.64
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14. 07.01.03: Construct the probability distribution for
a discrete random variable.
Example 7.2 Tossing coins
Values of a random variable
What is the probability distribution of the discrete random variable X that counts the number of
heads in four tosses of a coin? We can derive this distribution if we make two reasonable
assumptions:
1. The coin is balanced, so each toss is equally likely to give H or T.
2. The coin has no memory, so tosses are independent.
The outcome of four tosses is a sequence of heads and tails such as HTTH. There are 16 possible
outcomes in all. Figure 7.2 lists these outcomes along with the value of X for each outcome. The
multiplication rule for independent events tells us that, for example,
Each of the 16 possible outcomes similarly has probability 1/16. That is, these outcomes are equally
likely.
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23. Example 7.3 Random numbers and the uniform distribution
Areas under a density curve
The random number generator will spread its output uniformly across the entire interval from 0 to 1
as we allow it to generate a long sequence of numbers. The results of many trials are represented by
the density curve of a uniform distribution (Figure 7.5). This density curve has height 1 over the
interval from 0 to 1. The area under the density curve is 1, and the probability of any event is the
area under the density curve and above the event in question.
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28. Example 7.4 Cheating in school
Continuous random variables
Students are reluctant to report cheating by other students. A sample survey puts this question to an
SRS of 400 undergraduates: “You witness two students cheating on a quiz. Do you go to the
professor?” Suppose that if we could ask all undergraduates, 12% would answer “Yes.”
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