Incoming students to a certain school take a mathematics pla

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.

Solution

we\'ll express all the functions of x as fractions:

P( X=1) = 0

P (X=2) = 1/5

P (X=3) = 1/3

P(X=4) = 3/7

a)

P(Randomly selected student becomes a mathematics major)

= (0)*(10/100) + (1/5)*(20/100) + (1/3)*(60/100) + (3/7)*(10/100)

= 99/350

b) P(Randomly selected mathematics scores 4 given that she is a mathematics majors)

=  [(3/7)*(10/100)] /  [(0)*(10/100) + (1/5)*(20/100) + (1/3)*(60/100) + (3/7)*(10/100)]

= 5/33 [result of bayes theorem]

Hope this helps.