Answer:
The probability that a Randomly selected student from the incoming class will become a mathematics major is
or 0.2829
The probability that she scored a 4 on the placement exam is ![\frac{5}{33}\approx 0.1515](https://tex.z-dn.net/?f=%5Cfrac%7B5%7D%7B33%7D%5Capprox%200.1515)
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.
![P(M|x=1)=\frac{x-1}{x+3}=\frac{1-1}{1+3}=0](https://tex.z-dn.net/?f=P%28M%7Cx%3D1%29%3D%5Cfrac%7Bx-1%7D%7Bx%2B3%7D%3D%5Cfrac%7B1-1%7D%7B1%2B3%7D%3D0)
![P(M|x=2)=\frac{x-1}{x+3}=\frac{2-1}{2+3}=\frac{1}{5}](https://tex.z-dn.net/?f=P%28M%7Cx%3D2%29%3D%5Cfrac%7Bx-1%7D%7Bx%2B3%7D%3D%5Cfrac%7B2-1%7D%7B2%2B3%7D%3D%5Cfrac%7B1%7D%7B5%7D)
![P(M|x=3)=\frac{x-1}{x+3}=\frac{3-1}{3+3}=\frac{1}{3}](https://tex.z-dn.net/?f=P%28M%7Cx%3D3%29%3D%5Cfrac%7Bx-1%7D%7Bx%2B3%7D%3D%5Cfrac%7B3-1%7D%7B3%2B3%7D%3D%5Cfrac%7B1%7D%7B3%7D)
![P(M|x=4)=\frac{x-1}{x+3}=\frac{4-1}{4+3}=\frac{3}{7}](https://tex.z-dn.net/?f=P%28M%7Cx%3D4%29%3D%5Cfrac%7Bx-1%7D%7Bx%2B3%7D%3D%5Cfrac%7B4-1%7D%7B4%2B3%7D%3D%5Cfrac%7B3%7D%7B7%7D)
Part (A)
Now calculate the probability that a Randomly selected student from the incoming class will become a mathematics major.
![P(M)=\sum_{i=1}^{4}P(M|x_i)P(x_i)](https://tex.z-dn.net/?f=P%28M%29%3D%5Csum_%7Bi%3D1%7D%5E%7B4%7DP%28M%7Cx_i%29P%28x_i%29)
![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}](https://tex.z-dn.net/?f=P%3D%5Cfrac%7B1%7D%7B10%7D%5Ctimes%200%2B%5Cfrac%7B2%7D%7B10%7D%5Ctimes%20%5Cfrac%7B1%7D%7B5%7D%2B%5Cfrac%7B6%7D%7B10%7D%5Ctimes%20%5Cfrac%7B1%7D%7B3%7D%2B%5Cfrac%7B1%7D%7B10%7D%5Ctimes%20%5Cfrac%7B3%7D%7B7%7D)
![P=\frac{99}{350}\approx 0.2829](https://tex.z-dn.net/?f=P%3D%5Cfrac%7B99%7D%7B350%7D%5Capprox%200.2829)
Hence, the probability that a Randomly selected student from the incoming class will become a mathematics major is
or 0.2829
Part (B)
What is the probability that she scored a 4 on the placement exam?
![P(X_4|M)=\frac{P(M|x_4)P(x_4)}{P(M)}](https://tex.z-dn.net/?f=P%28X_4%7CM%29%3D%5Cfrac%7BP%28M%7Cx_4%29P%28x_4%29%7D%7BP%28M%29%7D)
![P(X_4|M)=\frac{\frac{3}{7}\cdot \frac{1}{10}}{\frac{99}{350}}](https://tex.z-dn.net/?f=P%28X_4%7CM%29%3D%5Cfrac%7B%5Cfrac%7B3%7D%7B7%7D%5Ccdot%20%5Cfrac%7B1%7D%7B10%7D%7D%7B%5Cfrac%7B99%7D%7B350%7D%7D)
![P=\frac{5}{33}\approx 0.1515](https://tex.z-dn.net/?f=P%3D%5Cfrac%7B5%7D%7B33%7D%5Capprox%200.1515)
Hence, the probability that she scored a 4 on the placement exam is ![\frac{5}{33}\approx 0.1515](https://tex.z-dn.net/?f=%5Cfrac%7B5%7D%7B33%7D%5Capprox%200.1515)