A worker has asked her supervisor for a letter of recommendation for a new job. She estimates that there is an 80 percent chance that she will get the job if she receives a strong recommendation, a 40 percent chance if she receives a moderately good recommendation, and a 10 percent chance if she receives a weak recommendation. She further estimates that the probabilities that the recommendation will be strong, moderate, and weak are .7, .2, and .1, respectively.

(a) How certain is she that she will receive the new job offer?

(b) Given that she does receive the offer, how likely should she feel that she received a strong recommendation? a moderate recommendation? a weak recommendation?

(c) Given that she does not receive the job offer, how likely should she feel that she received a strong recommendation? a moderate recommendation? a weak recommendation?

Given in the question that, there is an 80 percent chance for the worker to get the job if she receives a strong recommendation, a 40 percent chance if she receives a moderately good recommendation, and a 10 percent chance if she receives a weak recommendation.

We need to find how certain is she that she will receive the new job offer.

Consider $R$be the event that she receives a job offer

Sr : Strong Recommendation

Mr : Moderately good recommendations

Wr : Weak Recommendation

Here,

role="math" localid="1647073758970" $P\left(Sr\right)=0.7$

role="math" localid="1647073768545" $P\left(Mr\right)=0.2$

role="math" localid="1647073774395" $P\left(Wr\right)=0.1$

Let's take

$P\left(R\right|Mr)=40\%=0.4$

$P\left(R\right|Wr)=10\%=0.1$

The probability that she will receive the new job offer.

That is, $P\left(R\right)$

$P\left(R\right)=P\left(R\right|Sr)\times P(Sr)+P(R\left|Mr\right)\times P\left(Mr\right)+P\left(R\right|Wr)\times P(Wr)$

$0.7\times 0.8+0.2\times 0.4+0.1\times 0.1$

$0.56+0.08+0.01$

$0.65$

Therefore, the probability that she receives the new job offer would be$0.65$

the probability that she receives the new job offer would be$0.65$

Given in the question that, there is an 80 percent chance for the worker to get the job if she receives a strong recommendation, a 40 percent chance if she receives a moderately good recommendation, and a 10 percent chance if she receives a weak recommendation.

We need to find how likely should she feel that she received a strong recommendation? a moderate recommendation? a weak recommendation?

The probability that how likely should she feel that she received a strong recommendation, a moderate recommendation, a weak recommendation given that she does receive the job offer:

Strong recommendation

$P\left(Sr\right|R)=\frac{\left(R\right|Sr\left)P\right(Sr)}{P\left(R\right)}$

$P\left(Sr\right|R)=\frac{0.7\times 0.8}{0.65}$

$P\left(Sr\right|R)=0.8615$

Moderate Recommendation

$P\left(Mr\right|R)=\frac{P\left(R\right|Mr\left)P\right(Mr)}{P\left(R\right)}$

$P\left(Mr\right|R)=\frac{0.4\times 0.2}{0.65}$

$P\left(Mr\right|R)=0.12307=0.1231$

Weak Recommendation

$P\left(Wr\right|R)=\frac{P\left(R\right|Wr\left)P\right(Wr)}{P\left(R\right)}$

$P\left(Wr\right|R)=\frac{0.1\times 0.1}{0.65}$

$P\left(Wr\right|R)=0.01539=0.0154$

The probability that she feel that she received the job offer:

A strong recommendation $=0.8615$

A moderate Recommendation $=0.1231$

A weak Recommendation$0.0154$

Given in the question that, there is an 80 percent chance for the worker to get the job if she receives a strong recommendation, a 40 percent chance if she receives a moderately good recommendation, and a 10 percent chance if she receives a weak recommendation.

We need to find how certain is she that she will receive the new job offer.

From the information, she does not receive the job offer, the probability that how likely she should feel that she received a strong recommendation, a moderate recommendation, a weak recommendation.

$P\left(R^{\mathrm{\prime }}\right)=1-P\left(R\right)=1-0.65=0.35$

localid="1648017943472" role="math" $P(\left.{\mathrm{R}}^{\mathrm{\prime }}\mid \mathrm{Sr}\right)=1-0.8=0.2$

$P\left(R^{\mathrm{\prime }}\mid Mr\right)=1-0.4=0.6$

Therefore, The strong recommendation will be,

$P\left(Sr\mid R^{\mathrm{\prime }}\right)=\frac{P\left(R^{\mathrm{\prime }}\mid Sr\right)P\left(Sr\right)}{{\displaystyle P\left(R^{\mathrm{\prime }}\right)}}$

$P\left(Sr\mid R^{\mathrm{\prime }}\right)=\frac{0.2\times 0.7}{0.35}$

$P\left(Sr\mid R^{\mathrm{\prime }}\right)=0.40$

The moderate recommendation,

$P\left(Mr\mid R^{\mathrm{\prime }}\right)=\frac{P\left(R^{\mathrm{\prime }}\mid Mr\right)P\left(Mr\right)}{{\displaystyle P\left(R^{\mathrm{\prime }}\right)}}$

$P\left(Mr\mid R^{\mathrm{\prime }}\right)=\frac{0.6\times 0.2}{{\displaystyle 0.35}}$

$P\left(Mr\mid R^{\mathrm{\prime }}\right)=0.3429$

The Weak recommendation

$P\left(Wr\mid R^{\mathrm{\prime }}\right)=\frac{P\left(R^{\mathrm{\prime }}\mid Wr\right)P\left(Wr\right)}{P\left(R\right)}$

$P\left(Wr\mid R^{\mathrm{\prime }}\right)=\frac{0.9\times 0.1}{0.35}$

$P\left(Wr\mid R^{\mathrm{\prime }}\right)=0.2571$

If she does not receive the job offer, the probability that she should feel that she received:

a strong recommendation $0.40$

a moderate recommendations $0.3429$

a weak recommendation$0.2571$