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An instructor feels that 15% of students get below a C on their final exam. She decides to look at final exams (selected randomly and replaced in the pile after reading) until she finds one that shows a grade below a C. We want to know the probability that the instructor will have to examine at least ten exams until she finds one with a grade below a C. What is the probability question stated mathematically?
P ( x ≥ 10)
Suppose that you are looking for a student at your college who lives within five miles of you. You know that 55% of the 25,000 students do live within five miles of you. You randomly contact students from the college until one says he or she lives within five miles of you. What is the probability that you need to contact four people?
This is a geometric problem because you may have a number of failures before you have the one success you desire. Also, the probability of a success stays the same each time you ask a student if he or she lives within five miles of you. There is no definite number of trials (number of times you ask a student).
a. Let X = the number of ____________ you must ask ____________ one says yes.
a. Let
X = the number of
students you must ask
until one says yes.
b. What values does X take on?
b. 1, 2, 3, …, (total number of students)
c. What are p and q ?
c.
p = 0.55;
q = 0.45
d. The probability question is P (_______).
d. P ( x = 4)
You need to find a store that carries a special printer ink. You know that of the stores that carry printer ink, 10% of them carry the special ink. You randomly call each store until one has the ink you need. What are p and q ?
p = 0.1
q = 0.9
X ~ G ( p )
Read this as " X is a random variable with a geometric distribution ." The parameter is p ; p = the probability of a success for each trial.
The Geometric Pdf tells us the probability that the first occurrence of success requires x number of independent trials, each with success probability p. If the probability of success on each trial is p , then the probability that the x th trial (out of x trials) is the first success is:
for
x = 1, 2, 3, ....
The expected value of X, the mean of this distribution is 1/p. This tells us how many trials it will take to have a success including in the count the trial that results in success. The above form of the Geometric distribution is used for modeling the number of trials until the first success. The number of trials includes the one that is a success: x = all trials including the one that is a success. This can be seen in the form of the formula. If X = number of trials including the success, then we must multiply the probability of failure, (1-p), times the number of failures, that is X-1. By contrast, the following form of the geometric distribution is used for modeling number of failures until the first success:
for
x = 0, 1, 2, 3, ....
In this case the trial that is a success is not counted as a trial in the formula: x = number of failures. The expected value, mean, of this distribution is
. This tells us how many failures to expect before we have a success. In either case, the sequence of probabilities is a geometric sequence.
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