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

a

`P(G) =  0.69`

b

`P(S | G) = 0.81`

c

`P(M|G') =  0.26`

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) =  ( P(S) *  P(G|S))/( P(G))`

=>
`P(S|G) =  ( 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') =  ( P(M) *  (1- P(G|M)))/((1 - P(G)))`

`P(M| G') =  ( 0.2 *  (1- 0.6))/( (1 - 0.69))`

`P(M|G') =  0.26`