With explanation please You want to go to graduate school, so you ask your math

Question

With explanation please

You want to go to graduate school, so you ask your math professor, Dr. Emmy Noether, for a letter of recommendation. You estimate that there is a 80% chance that you will get into a graduate program if you receive a strong recommendation, a 60% chance that you will get into a graduate program if you receive a moderately good recommendation, and 5% chance that you will get into a graduate program if you receive a weak recommendation. Furthermore, you estimate that the probabilities that a recommendation will be strong, moderately good, and weak are 0.7, 0.2, and 0.1, respectively. Based on these estimates, what is the probability that you will get into a graduate program. Given that you did receive an offer to attend a graduate program, what is the probability that you received a strong recommendation? Suppose you didn't receive an offer to attend a graduate program. Given that, what is the probability that you received a moderately good recommendation?

Explanation / Answer

let probabilty of strong recommendation =P(S) ; moderate =P(M) and weak =P(W)

and entering into graduate programme =P(G)

here P(S) =0.7 ; P(M) =0.2 ; P(W) =0.1

a) hence from bayes theorum ; probabilty to get into a graduate programme P(G)

=P(S)*P(G|S) +P(M)*P(G|M) +P(W)*P(G|W)

=0.7*0.8+0.2*0.6+0.1*0.05 =0.685

b) probabilty you get a strong recommendation, given you receive offer to get in to a graduate programme

=P(S|G) =P(S)*P(G|S)/P(G) =0.7*0.8/0.685 =0.8175

c) probabilty of not receiving an offfer to attend a graduate programme =P(G') =1-P(G) =1-0.685 =0.315

probabilty you get a moderate recommendation, given you did not receive offer to get in to a graduate programme

=P(M|G') =P(M)*P(G'|M)/P(G') =0.2*(1-0.6)/0.315 =0.254

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