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|0.163|\n|2|0.119|\n|3|0.16|\n|4|0.061|\n|5|0.013|\n|6|0.01|\n|7|0.003|\n|8|0.002|\na. what is the probability that a randomly selected student changed his or her major at least once?\nb. what is the probability that a randomly selected student changed his or her major at most twice?\nc. 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|0.163|\n|2|0.119|\n|3|0.16|\n|4|0.061|\n|5|0.013|\n|6|0.01|\n|7|0.003|\n|8|0.002|\na. what is the probability that a randomly selected student changed his or her major at least once?\nb. what is the probability that a randomly selected student changed his or her major at most twice?\nc. 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: Define the probability - sum rule

The sum of all probabilities in a probability distribution is 1. For a discrete random variable (X) with probabilities (P(X = k)) for (k = 0,1,\cdots,n), (\sum_{k}P(X = k)=1).

Step2: Calculate the probability for part a

The probability that a randomly - selected student changed his or her major at least once is (P(X\geq1)). Using the complement rule (P(A)=1 - P(\text{not }A)), we have (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 randomly - selected student changed his or her major at most twice is (P(X\leq2)). We sum the probabilities for (X = 0), (X = 1), and (X = 2). So (P(X\leq2)=P(X = 0)+P(X = 1)+P(X = 2)=0.247 + 0.163+0.119 = 0.529).

Step4: Calculate the probability for part c

The probability that a randomly - selected person changed majors more than three times is (P(X>3)). We sum the probabilities for (X = 4), (X = 5), (X = 6), (X = 7), and (X = 8). (P(X>3)=P(X = 4)+P(X = 5)+P(X = 6)+P(X = 7)+P(X = 8)=0.061+0.013+0.01+0.003+0.002 = 0.09).

Answer:

a. (0.753) b. (0.529) c. (0.09)