Question 1135875
<font face="times" color="black" size="3">Part A


The events are not mutually exclusive because there are 6 females who like Five Guys Burgers best. If these two events were mutually exclusive, then this value would be 0. Mutually exclusive events are two such events where they cannot happen at the same time. 


An example of mutual exclusive events would be the events "heads" and "tails" on some coin. You can only have one event happen, but not both at the same time.
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Part B


Focus only on the male column. We have 248 total and 162 who like In-N-Out Burger best, so the probability is therefore,


162/248 = 0.65322580645161 = <font color=red size=4>0.6532</font>


It appears you have the right idea but you only put the answer accurate to two decimal places (rather than four). 
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Part C


This time we only focus on the female column
181 females like In-N-Out Burger best
252 females total


181/252 = 0.71825396825397 = <font color=red size=4>0.7183</font>
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Part D


Define the two events
M = event that a male likes In-N-Out Burger best
N = event that a female likes In-N-Out Burger best


From the previous parts (B and C), we found that
P(M) = probability male likes In-N-Out best
P(M) = 0.6532
and
P(N) = probability female likes In-N-Out best
P(N) = 0.7183


Assuming the events M and N are independent, this would allow us to multiply the probability values
P(M and N) = P(M)*P(N)
P(M and N) = 0.6532*0.7183
P(M and N) = 0.46919356
P(M and N) = <font color=red size=4>0.4692</font>


The key thing is that the formula only works if M and N are independent. Each person surveyed is likely to not worry about what other people are doing or thinking, since they are probably alone taking it. So no single person affects any other in terms of whether they like a certain burger place. Also, a person's tastes are unique to them so even if others are present, they aren't likely to sway an opinion. The survey has a subjective base to it (ie there is no right or wrong answer). 


So in short, the assumption needed to be made is that the events are independent. It's is reasonable to assume this. 

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Part E


252 females
6 people like Umami Burger best
1 person is both female and they like Umami Burger best
252+6-1 = 257 people are either female, like Umami Burger best, or both apply


note: you subtract off that 1 person because they are in both categories, so effectively this one person is counted twice when we added the 252 and 6 together


There are 257 people we want out of 500 total, so the probability is:
257/500= <font color=red size=4>0.5140</font>
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