Geometric Distribution Quiz

PDF page 267: There are three main characteristics of a geometric experiment.

PDF page 267: 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.

PDF page 267: 1 For example, the probability of rolling a three when you throw one fair die is . This is true no matter how many 6 times you roll the die. Suppose you want to know the probability of getting the first three on the fifth roll. On rolls 5 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 6 ⎛5⎞⎛5⎞⎛5⎞⎛5⎞⎛1⎞ = 0.0804 failure. The probability of getting a three on the fifth roll is ⎝6⎠⎝6⎠⎝6⎠⎝6⎠⎝6⎠ X = the number of independent trials until the first success.

PDF page 267: 1 For example, the probability of rolling a three when you throw one fair die is . This is true no matter how many 6 times you roll the die. Suppose you want to know the probability of getting the first three on the fifth roll. On rolls 5 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 6 ⎛5⎞⎛5⎞⎛5⎞⎛5⎞⎛1⎞ = 0.0804 failure. The probability of getting a three on the fifth roll is ⎝6⎠⎝6⎠⎝6⎠⎝6⎠⎝6⎠ X = the number of independent trials until the first success.

PDF page 267: 1 For example, the probability of rolling a three when you throw one fair die is . This is true no matter how many 6 times you roll the die. Suppose you want to know the probability of getting the first three on the fifth roll. On rolls 5 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 6 ⎛5⎞⎛5⎞⎛5⎞⎛5⎞⎛1⎞ = 0.0804 failure. The probability of getting a three on the fifth roll is ⎝6⎠⎝6⎠⎝6⎠⎝6⎠⎝6⎠ X = the number of independent trials until the first success.

PDF page 268: 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).

PDF page 268: 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).

PDF page 268: 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).

PDF page 268: 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).

PDF page 268: 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).

PDF page 269: b. 1, 2, 3, ..., (total number of students)

PDF page 269: c. p = 0.55; q = 0.45