Question 1203260
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Answer: <font color=red size=4>Choice C) 0.240</font>


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Explanation


There are at least two methods we can use. 


Method 1


A = Alonzo buses tables in the middle area
B = Bob buses tables in the middle area
C = Casper buses tables in the middle area
D = Dish is broken in the middle area
 

From the second sentence of the instructions, we are given these 3 pieces of info:
P(A) = 0.60
P(B) = 0.10
P(C) = 0.30


The next 3 sentences offer these additional 3 pieces of info:
P(D given A) = 0.06
P(D given B) = 0.02
P(D given C) = 0.04


<font color=red>Goal: Compute P(C given D)</font>
In other words, we want to find P(C) when we know event D has occurred. 


Use <a href="https://en.wikipedia.org/wiki/Bayes%27_theorem">Bayes' Theorem</a>
P(C given D) = P(D given C)*P(C)/P(D)


We already have the values of P(D given C) and P(C)


We'll need P(D)


Use the <a href="https://en.wikipedia.org/wiki/Law_of_total_probability">Law of Total Probability</a>
P(D) = P(D and A) + P(D and B) + P(D and C)
P(D) = P(D given A)*P(A) + P(D given B)*P(B) + P(D given C)*P(C)
P(D) = 0.06*0.60 + 0.02*0.10 + 0.04*0.30
P(D) = 0.05


We have enough info to compute the following
P(C given D) = P(D given C)*P(C)/P(D)
P(C given D) = 0.04*0.30/0.05
P(C given D) = 0.012/0.05
P(C given D) = 12/50
P(C given D) = 6/25
P(C given D) = <font color=red size=4>0.240</font>


This is why the final answer is <font color=red size=4>choice C</font>


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Method 2


Let's say that there are 1000 instances of a single dish being bused.


Alonzo handles 60% of them, Bob does 10%, Casper does the remaining 30%
60% of 1000 = 0.60*1000 = 600
10% of 1000 = 0.10*1000 = 100
30% of 1000 = 0.30*1000 = 300


So,
Alonzo buses 600 times
Bob buses 100 times
Casper buses 300 times


Alonzo breaks a dish 6% of the time
6% of 600 = 0.06*600 = 36
If Alonzo buses 600 times then we expect him to break around 36 dishes.


Bob breaks a dish 2% of the time
2% of 100 = 0.02*100 = 2
If Bob buses 100 times then we expect him to break around 2 dishes.


Casper breaks a dish 4% of the time
4% of 300 = 0.04*300 = 12
If Casper buses 300 times then we expect him to break around 12 dishes.


An total estimate of 36+2+12 = 50 dishes have been broken.
P(D) = probability of a broken dish
P(D) = (50 dishes broken)/(1000 dishes total)
P(D) = 50/1000
P(D) = 0.05
This was calculated earlier in the previous section.


Of those 50 broken dishes, Casper is estimated to break 12 of them.
12/50 = <font color=red>0.240</font>
If we encounter a broken dish, then there's a 24% chance that Casper did it.
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