2. (25 pts.) we have three masons in our construction site. lets call them good mason, average mason and bad…

2. (25 pts.) we have three masons in our construction site. lets call them good mason, average mason and bad mason. the probability that good mason builds a wall that is fully vertical is 90%. this probability is 60% for average mason. for bad mason, the probability is 20% on cloudy days and 50% in sunny days. bad mason builds half of the walls in the construction site. good mason and average mason each build half of the rest of the walls. on average, 70% of the days during the construction season are sunny. weather does not affect who builds the walls. (a) yesterday was cloudy, and on that day, someone built a wall. what is the probability that the wall is fully vertical? (b) someone built a fully vertical wall the day before yesterday. what is the probability that the day before yesterday was sunny?

2. (25 pts.) we have three masons in our construction site. lets call them good mason, average mason and bad mason. the probability that good mason builds a wall that is fully vertical is 90%. this probability is 60% for average mason. for bad mason, the probability is 20% on cloudy days and 50% in sunny days. bad mason builds half of the walls in the construction site. good mason and average mason each build half of the rest of the walls. on average, 70% of the days during the construction season are sunny. weather does not affect who builds the walls. (a) yesterday was cloudy, and on that day, someone built a wall. what is the probability that the wall is fully vertical? (b) someone built a fully vertical wall the day before yesterday. what is the probability that the day before yesterday was sunny?

Answer

Explanation:

Step1: Define probabilities

Let $G$, $A$, $B$ be the events that the wall is built by the good - mason, average - mason, and bad - mason respectively. Since each mason builds half of the walls, $P(G)=P(A)=P(B)=\frac{1}{3}$. Let $V$ be the event that the wall is vertical. Let $C$ be the event that it is cloudy and $S$ be the event that it is sunny, with $P(S) = 0.7$ and $P(C)=0.3$. The conditional probabilities are: $P(V|G)=0.9$, $P(V|A)=0.6$, $P(V|B,C)=0.2$, $P(V|B,S)=0.5$.

Step2: Calculate $P(V)$ for part (a)

By the law of total probability, $P(V)=P(V|G)P(G)+P(V|A)P(A)+P(V|B,C)P(B)P(C)+P(V|B,S)P(B)P(S)$. Substitute the values: [ \begin{align*} P(V)&=0.9\times\frac{1}{3}+0.6\times\frac{1}{3}+0.2\times\frac{1}{3}\times0.3 + 0.5\times\frac{1}{3}\times0.7\ &=\frac{0.9}{3}+\frac{0.6}{3}+\frac{0.2\times0.3}{3}+\frac{0.5\times0.7}{3}\ &=\frac{0.9 + 0.6+0.06 + 0.35}{3}\ &=\frac{1.91}{3}\approx0.637 \end{align*} ]

Step3: Use Bayes' theorem for part (b)

First, find $P(V)$ using the law of total probability: [ \begin{align*} P(V)&=P(V|G)P(G)+P(V|A)P(A)+P(V|B,C)P(B)P(C)+P(V|B,S)P(B)P(S)\ &=0.9\times\frac{1}{3}+0.6\times\frac{1}{3}+0.2\times\frac{1}{3}\times0.3+0.5\times\frac{1}{3}\times0.7\ &=\frac{0.9 + 0.6+0.06+0.35}{3}=\frac{1.91}{3} \end{align*} ] By Bayes' theorem, $P(S|V)=\frac{P(V|S)P(S)}{P(V)}$. We calculate $P(V|S)=P(V|G)P(G)+P(V|A)P(A)+P(V|B,S)P(B)$ [ \begin{align*} P(V|S)&=0.9\times\frac{1}{3}+0.6\times\frac{1}{3}+0.5\times\frac{1}{3}\ &=\frac{0.9 + 0.6+0.5}{3}=\frac{2}{3} \end{align*} ] Then $P(S|V)=\frac{P(V|S)P(S)}{P(V)}=\frac{\frac{2}{3}\times0.7}{\frac{1.91}{3}}=\frac{1.4}{1.91}\approx0.733$

Answer:

(a) $\frac{1.91}{3}\approx0.637$ (b) $\frac{1.4}{1.91}\approx0.733$