data were collected from a survey given to graduating college seniors on the number of times they had…

data were collected from a survey given to graduating college seniors on the number of times they had changed majors. from that data, a probability distribution was constructed. the random variable x is defined as the number of times a graduating senior changed majors. it is shown below:\n| x | p(x = x) |\n|----|----| \n| 0 | 0.247 |\n| 1 | 0.165 |\n| 2 | 0.319 |\n| 3 | 0.16 |\n| 4 | 0.061 |\n| 5 | 0.033 |\n| 6 | 0.01 |\n| 7 | 0.003 |\n| 8 | 0.002 |\n a. what is the probability that a randomly selected student changed his or her major at least once? \n b. what is the probability that a randomly selected student changed his or her major at most twice? \n c. given that a randomly selected person did change majors, what is the probability that he or she changed majors more than three times?

data were collected from a survey given to graduating college seniors on the number of times they had changed majors. from that data, a probability distribution was constructed. the random variable x is defined as the number of times a graduating senior changed majors. it is shown below:\n| x | p(x = x) |\n|----|----| \n| 0 | 0.247 |\n| 1 | 0.165 |\n| 2 | 0.319 |\n| 3 | 0.16 |\n| 4 | 0.061 |\n| 5 | 0.033 |\n| 6 | 0.01 |\n| 7 | 0.003 |\n| 8 | 0.002 |\n a. what is the probability that a randomly selected student changed his or her major at least once? \n b. what is the probability that a randomly selected student changed his or her major at most twice? \n c. given that a randomly selected person did change majors, what is the probability that he or she changed majors more than three times?

Answer

Explanation:

Step1: Recall probability - sum rule

The sum of all probabilities in a probability - distribution is 1. For a discrete random variable (X) with probability mass function (P(X = x)), (\sum_{x}P(X = x)=1).

Step2: Calculate the probability for part a

The probability that a student changed his or her major at least once is (P(X\geq1)). We know that (P(X\geq1)=1 - P(X = 0)). Given (P(X = 0)=0.247), then (P(X\geq1)=1 - 0.247 = 0.753).

Step3: Calculate the probability for part b

The probability that a student changed his or her major at most twice is (P(X\leq2)). We use the formula (P(X\leq2)=P(X = 0)+P(X = 1)+P(X = 2)). Given (P(X = 0)=0.247), (P(X = 1)=0.165), and (P(X = 2)=0.319), then (P(X\leq2)=0.247 + 0.165+0.319 = 0.731).

Step4: Calculate the probability for part c

The probability that a student changed his or her major more than three times is (P(X>3)). We use the formula (P(X>3)=P(X = 4)+P(X = 5)+P(X = 6)+P(X = 7)+P(X = 8)). Given (P(X = 4)=0.061), (P(X = 5)=0.033), (P(X = 6)=0.01), (P(X = 7)=0.003), (P(X = 8)=0.002), then (P(X>3)=0.061 + 0.033+0.01+0.003+0.002=0.11).

Answer:

a. (0.753) b. (0.731) c. (0.11)