Question

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 1

Answer:

The probability that a Randomly selected student from the incoming class will become a mathematics major is $\frac{99}{350}$ or 0.2829

The probability that she scored a 4 on the placement exam is $\frac{5}{33}\approx 0.1515$

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.

$P(M|x=1)=\frac{x-1}{x+3}=\frac{1-1}{1+3}=0$

$P(M|x=2)=\frac{x-1}{x+3}=\frac{2-1}{2+3}=\frac{1}{5}$

$P(M|x=3)=\frac{x-1}{x+3}=\frac{3-1}{3+3}=\frac{1}{3}$

$P(M|x=4)=\frac{x-1}{x+3}=\frac{4-1}{4+3}=\frac{3}{7}$

Part (A)

Now calculate the probability that a Randomly selected student from the incoming class will become a mathematics major.

$P(M)=\sum_{i=1}^{4}P(M|x_i)P(x_i)$

$P=\frac{1}{10}\times 0+\frac{2}{10}\times \frac{1}{5}+\frac{6}{10}\times \frac{1}{3}+\frac{1}{10}\times \frac{3}{7}$

$P=\frac{99}{350}\approx 0.2829$

Hence, the probability that a Randomly selected student from the incoming class will become a mathematics major is $\frac{99}{350}$ or 0.2829

Part (B)

What is the probability that she scored a 4 on the placement exam?

$P(X_4|M)=\frac{P(M|x_4)P(x_4)}{P(M)}$

$P(X_4|M)=\frac{\frac{3}{7}\cdot \frac{1}{10}}{\frac{99}{350}}$

$P=\frac{5}{33}\approx 0.1515$

Hence, the probability that she scored a 4 on the placement exam is $\frac{5}{33}\approx 0.1515$