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	0	1	2	3	4	5	6	7	8
p(x = x)	0.212	0.319	0.162	0.166	0.089	0.037	0.011	0.003	0.001

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: Recall probability formula

The probability of an event $A$ is $P(A)$. For complementary events, $P(A)=1 - P(\text{not }A)$.

Step2: Calculate probability for part a

The probability of changing major at least once is the complement of changing major 0 times. Let $A$ be the event of changing major at least once. Then $P(A)=1 - P(X = 0)$. Given $P(X = 0)=0.212$, so $P(A)=1 - 0.212 = 0.788$.

Step3: Calculate probability for part b

The probability of changing major at most twice is $P(X\leq2)=P(X = 0)+P(X = 1)+P(X = 2)$. Given $P(X = 0)=0.212$, $P(X = 1)=0.319$, $P(X = 2)=0.162$. Then $P(X\leq2)=0.212 + 0.319+0.162=0.693$.

Step4: Calculate probability for part c

Let $B$ be the event of changing majors and $C$ be the event of changing majors more than three times. First, $P(B)=1 - P(X = 0)=0.788$. $P(C)=P(X = 4)+P(X = 5)+P(X = 6)+P(X = 7)+P(X = 8)=0.089 + 0.037+0.011+0.003+0.001=0.141$. By the formula for conditional - probability $P(C|B)=\frac{P(B\cap C)}{P(B)}$, and since $B\cap C = C$ in this case, $P(C|B)=\frac{0.141}{0.788}\approx0.179$.

Answer:

a. $0.788$
b. $0.693$
c. $0.179$