Respuesta :
Answer:
The probability that a Randomly selected student from the incoming class will become a mathematics major is [tex]\frac{99}{350}[/tex] or 0.2829
The 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=1/10
20% of the students scored a 2 = 20% = 20/100=2/10
60% of the students scored a 3 = 60% = 60/100=6/10
10% of the students scored a 4 = 10% = 10/100=1/10
The student will become a mathematics major with probability x-1/x+3.
Calculate the probability for x=1,2,3 and 4
Let 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 0.2829
Part (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]
