Question

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?

Answer 1

Answer:

a

  $P(G) =  0.69$

b

  $P(S | G) = 0.81$

c

  $P(M|G') =  0.26$

Step-by-step explanation:

From the question we are told the

   The probability of getting into getting into graduated school if you receive a strong recommendation is  $P(G |S) = 0.80$

   The probability of getting into getting into graduated school if you receive a moderately good recommendation is  $P(G| M) =  0.60$

   The probability of getting into getting into graduated school if you receive a weak recommendation is  $P(G|W) =  0.05$

   The probability of getting a strong recommendation is  $P(S) =  0.7$

     The  probability of receiving a moderately good recommendation is $P(M) =  0.2$

       The probability of receiving a weak recommendation is $P(W) =  0.1$

      Generally  the probability that you will get into a graduate program is mathematically represented as

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

=>   $P(G) =  0.7 * 0.8 +  0.2 *  0.6 + 0.1 *  0.05$

=>   $P(G) =  0.69$

Generally  given that you did receive an offer to attend a graduate program, what is the probability that you received a strong recommendation is mathematically represented as

      $P( S|G) =  \frac{ P(S) *  P(G|S)}{ P(G)}$

=>    $P(S|G) =  \frac{ 0.7 * 0.8 }{0.69}$

=>     $P(S | G) = 0.81$

Generally given that you didn't receive an offer to attend a graduate program  the probability that you received a moderately good recommendation is mathematically represented as

        $P(M|G') =  \frac{ P(M) *  (1- P(G|M))}{(1 - P(G))}$

         $P(M| G') =  \frac{ 0.2 *  (1- 0.6)}{ (1 - 0.69)}$

         $P(M|G') =  0.26$