Question

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:

x	p(x = x)
1	0.163
2	0.119
3	0.16
4	0.061
5	0.013
6	0.01
7	0.003
8	0.002

a. what is the probability that a randomly selected student changed his or her major at least once?
b. what is the probability that a randomly selected student changed his or her major at most twice?
c. given that a randomly selected person did change majors, what is the probability that he or she changed majors more than three times?

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. \(0.09\)