A community college classifies a student as a freshman or sophomore depending on the number ofcredits that the student has earned. Data from the school indicate that from one fall semester to thenext 25% of freshman will graduate, 50% will become sophomores, and 25% will quit. Furthermore, 50%of sophomores will graduate, 25% will remain sophomores, and 25% will quit. Use the techniques fromlecture 11 to determine theprobabilitythat an incoming freshman will graduate.

P(A freshman will graduate) = 1 - 0.4167 = 0.5833

Explanation

Probability that a freshman will graduate = 0.25

Probability that a freshman will become sophomores = 50% = 0.50

Probability that he will quit = 25% =0.25

Now.

Probability that a sophomore will graduate = 0.50

Probability that a sophomore will remain sophomore = 0.25

Probability that it will quit = 0.25

so,

P(Freshman will graduate) = P(WHo directly graduated from freshman) + P(WHo graduated by becoming sophomore first and then graduated)

Here,

P(Who directly graduated from freshman) = 0.25

P(Who graduated by becoming sophomore first and then graduated) = P(Freshman -> Sophomore -> Graduate) + P(Freshman -> Sophomore -> Sophomore -> Graduate) + P(Freshman -> Sophomore -> Sophomore -> Sophomore -> Graduate) +

= 0.25 + 0.5 * [0.5 + 0.25 * 0.5 + 0.252 * 0.5 + ...]

= 0.25 + 0.52 * 1/(1 - 0.25)

= 0.25 + 0.52/0.75

= 0.5833

P(Freshman who will graduated) = 0.5833

Method II

P(A freshman will graduate) = 1 - P(A freshman that will quit)

P(Freshman will quit) = P(Quit when he is a freshman) + P(Quit when he become sophomore from a freshman)

= 0.25 + [0.50 * 0.25 + 0.50 * 0.50 * 0.25 + ...]

= 0.25 + 0.50 * 0.25 * 1/(1 - 0.25)

= 0.25 + 0.50 * 0.25 * 1/0.75

= 0.4167

so

P(A freshman will graduate) = 1 - 0.4167 = 0.5833