There are three main characteristics of a geometric experiment.
There are one or more Bernoulli trials with all failures except the last one, which is a success. In other words, you keep repeating what you are doing until the first success. Then you stop. For example, you throw a dart at a bullseye until you hit the bullseye. The first time you hit the bullseye is a "success" so you stop throwing the dart. It might take six tries until you hit the bullseye. You can think of the trials as failure, failure, failure, failure, failure, success, STOP.
In theory, the number of trials could go on forever. There must be at least one trial.
The probability,
p , of a success and the probability,
q , of a failure is the same for each trial.
p +
q = 1 and
q = 1 −
p . For example, the probability of rolling a three when you throw one fair die is
. This is true no matter how many times you roll the die. Suppose you want to know the probability of getting the first three on the fifth roll. On rolls one through four, you do not get a face with a three. The probability for each of the rolls is
q =
, the probability of a failure. The probability of getting a three on the fifth roll is
= 0.0804
X = the number of independent trials until the first success.
You play a game of chance that you can either win or lose (there are no other possibilities)
until you lose. Your probability of losing is
p = 0.57. What is the
probability that it takes five games until you lose? Let
X = the number of games you play until you lose (includes the losing game). Then
X takes on the values 1, 2, 3, ... (could go on indefinitely). The probability question is
P (
x = 5).
You throw darts at a board until you hit the center area. Your probability of hitting the center area is
p = 0.17. You want to find the probability that it takes eight throws until you hit the center. What values does
X take on?
A safety engineer feels that 35% of all industrial accidents in her plant are caused by failure of employees to follow instructions. She decides to look at the accident reports (selected randomly and replaced in the pile after reading)
until she finds one that shows an accident caused by failure of employees to follow instructions. On average, how many reports would the safety engineer
expect to look at until she finds a report showing an accident caused by employee failure to follow instructions? What is the probability that the safety engineer will have to examine at least three reports until she finds a report showing an accident caused by employee failure to follow instructions?
Let
X = the number of accidents the safety engineer must examine
until she finds a report showing an accident caused by employee failure to follow instructions.
X takes on the values 1, 2, 3, .... The first question asks you to find the
expected value or the mean. The second question asks you to find
P (
x ≥ 3). ("At least" translates to a "greater than or equal to" symbol).
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?
Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ...
Step 2: Find each score's deviation from the mean. ...
Step 3: Square each deviation from the mean. ...
Step 4: Find the sum of squares. ...
Step 5: Divide the sum of squares by n – 1 or N.
The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.
Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400
a. what is the probability of getting more than 12,000 hits?
b. what is the probability of getting fewer than 9,000 hits?
Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400.
a. What is the probability of getting more than 12,000 hits
