Incoming students to a certain school take a mathematics placement exam. The possible scores are 1, 2, 3, and 4. From past experience, the school knows that if a particular student's score is x {1,2,3,4}, then the student will become a mathematics major with probability x-1/x+3 Suppose that the incoming class had the following scores: 10% of the students scored a 1, 20% scored a 2, 60% scored a 3, and 10% scored a 4. What is the probability that a Randomly selected student from the incoming class will become a mathematics major? Express your answer as a fraction in lowest terms. Suppose a randomly selected student from the incoming class turns out to be a mathematics major. What is the probability that she scored a 4 on the placement exam? Express your answer as a fraction in lowest terms.

Answer:The probability that a Randomly selected student from the incoming class will become a mathematics major is [tex]\frac{99}{350}[/tex] or 0.2829The probability that she scored a 4 on the placement exam is [tex]\frac{5}{33}\approx 0.1515[/tex]Step-by-step explanation:Consider the provided information.Then, the given student score is:10% of the students scored a 1 = 10% = 10/100=1/1020% of the students scored a 2 = 20% = 20/100=2/1060% of the students scored a 3 = 60% = 60/100=6/1010% of the students scored a 4 = 10% = 10/100=1/10The student will become a mathematics major with probability x-1/x+3.Calculate the probability for x=1,2,3 and 4Let the event  M  denote that a randomly selected student will become a math major.[tex]P(M|x=1)=\frac{x-1}{x+3}=\frac{1-1}{1+3}=0[/tex][tex]P(M|x=2)=\frac{x-1}{x+3}=\frac{2-1}{2+3}=\frac{1}{5}[/tex][tex]P(M|x=3)=\frac{x-1}{x+3}=\frac{3-1}{3+3}=\frac{1}{3}[/tex][tex]P(M|x=4)=\frac{x-1}{x+3}=\frac{4-1}{4+3}=\frac{3}{7}[/tex]Part (A)Now calculate the probability that a Randomly selected student from the incoming class will become a mathematics major.[tex]P(M)=\sum_{i=1}^{4}P(M|x_i)P(x_i)[/tex][tex]P=\frac{1}{10}\times 0+\frac{2}{10}\times \frac{1}{5}+\frac{6}{10}\times \frac{1}{3}+\frac{1}{10}\times \frac{3}{7}[/tex][tex]P=\frac{99}{350}\approx 0.2829[/tex]Hence, the probability that a Randomly selected student from the incoming class will become a mathematics major is [tex]\frac{99}{350}[/tex] or 0.2829Part (B)What is the probability that she scored a 4 on the placement exam?[tex]P(X_4|M)=\frac{P(M|x_4)P(x_4)}{P(M)}[/tex][tex]P(X_4|M)=\frac{\frac{3}{7}\cdot \frac{1}{10}}{\frac{99}{350}}[/tex][tex]P=\frac{5}{33}\approx 0.1515[/tex]Hence, the probability that she scored a 4 on the placement exam is [tex]\frac{5}{33}\approx 0.1515[/tex]