GMAT Prep

[Jameschan168]

A certain manufacturer of cake, muffin, and bread mixes has 100 buyers, of whom 50 purchase cake mix, 40 purchase muffin mix, and 20 purchase both cake mix and muffin mix. If a buyer is to be selected at random from the 100 buyers, what is the probability that the buyer selected will be one who purchase neither cake mix or muffin mix?

a. 1/10
b. 3/10
c. 1/2
d. 7/10
e. 9/10

[ri2007]

A certain manufacturer of cake, muffin, and bread mixes has 100 buyers, of whom 50 purchase cake mix, 40 purchase muffin mix, and 20 purchase both cake mix and muffin mix. If a buyer is to be selected at random from the 100 buyers, what is the probability that the buyer selected will be one who purchase neither cake mix or muffin mix?

a. 1/10
b. 3/10
c. 1/2
d. 7/10
e. 9/10

Ans should be 1/10

Total sample size = 100
Total no. people who buy neither Cake or Muffing mix =10

So ans is 10/100 = 1/10

Pls let me know if i am missing some thing

[samirpandeyit62]

Yes RI it should be 1/10 only as 20 is overlapped & thus included within the set of 50 & 40 so 10 are left.

[optimisticsam]

I must be missing something. This is how I solved it (perhaps wrongly) -
50 cake minus 20 who purchased both = 30 just purchased cake
40 muffin minus the 20 who got both = 20 purchased just muffin mix

So 50 purchased either muffin OR cake and 20 purchased both. I.E. 70 different people.

Therefore, 100-70=30 didn't purchase either.... so 3/10 is answer?

What am I screwing up here? ANything?

[samirpandeyit62]

Hi optimisticsam,
In this Q I believe that the data of 20v(c&m) is only given to confuse people, it does not hold any value in solving the Q, so one must not consider it, in my post I mentioned the operlap, i.e the data of 20 will be counted for 50 & 40, you wrote:
50 cake minus 20 who purchased both = 30 just purchased cake
40 muffin minus the 20 who got both = 20 purchased just muffin mix

now here it is valid to subtract this figure of 20 from eithet 50 or 40 as it counted in both of them, hence it must subrscted from 50+40, to remove this double counting, but as a matter of fact all this does't matter at all here as we need only data of 50+40 i.e 90 which would anyway include this overlap i.e figure of 20

we can say this 50 + 40 includes people who brought C or M, (including the people who brought both) that is what is required.

[optimisticsam]

I am still confused a little.

I think it would be valid to include this data because we are looking for the probability that someone selected would be someone who purchased neither. Therefore, couldn't we make a ven diagram with 30 cake - 20 muffin - 20 in the middle with both. Thus leaving 30 outside the diagram?

Wouldn't that mean the answer should be 3/10?

Is a ven diagram not the correct way to solve this?

Sorry if I am way off base, just trying to be clear on this.

Thanks Samir

[samirpandeyit62]

Hi optimisticsam,
Yes u are correct the ans should be 3/10, now in set theory we have a standard rule that

A Union B = A + B - A Intersect B

Now A Union B is also called A Or B & A intersect B is called A and B

If you interpret this in English, A Or B means either A or B but not both
(This is were the confusion was)
As per set theory this should mean A or B or BOTH,

hence in the problem we have 50+40-20 =70 is the nos of buyers who brought both cake or muffin or BOTH, hence 100-70 =30 buyers who did not buy both

so the probabilty is 3/10

Thanks for correcting us.

[ri2007]

ahhh when will i learn to avoid simple mistakes!!!

thanks optimisticsam & Samir

[samirpandeyit62]

Hi All,
these simple mistakes mean the difference between 51 Q and 48 Q, & everybody is susceptible to them. However I stongly feel that if one'e approach is right & concepts transparently clear then 51 Q is not a big thing to achieve. Would u guys like to add anything else to this.

One more important thing ,I recollect why I developed this confusion, It is because of the following, I would really appreciate if u guys or anyone give me your comments on this.

How will you interpret the following two statements

stmt 1 : The nos of buyers who buy cake (C) or muffin (M) is 50

Acoording to me this should be C Or M

i.e C + M -C and M (people who buy both)

stmt 2: The nos of buyers who buy either cake or muffin is 50

Now does this mean C + M - 2(C and M) (people who buy both)

I presumed this similar to stmt 1 which got me a question wrong during my practice.

Really appreciate your comments

[gabriel]

Hi optimisticsam,
Yes u are correct the ans should be 3/10, now in set theory we have a standard rule that

A Union B = A + B - A Intersect B

Now A Union B is also called A Or B & A intersect B is called A and B

If you interpret this in English, A Or B means either A or B but not both
(This is were the confusion was)
As per set theory this should mean A or B or BOTH,

hence in the problem we have 50+40-20 =70 is the nos of buyers who brought both cake or muffin or BOTH, hence 100-70 =30 buyers who did not buy both

so the probabilty is 3/10

Thanks for correcting us.

this would be true only if there were 2 elements(only cake and muffin mix) to the question .. but u seem to be forgetting that there is a 3rd element to this question, bread mix.. so the answer( or is it the question) doesnt seem to be right ... any comments ?? ..

[samirpandeyit62]

Hi Gabriel,
50 + 40 - 20 =70

this figure represents the nos of buyers who buy either cake or muffin or both

so 100 -70 =30 would be the nos of buyers left to buy anything from except cake or muffin, which is bread only. Ya if there were a couple of more poducts then, then we could not have made such a statement.

now since there are only 3 products & these 30 guys are buyers who buy something other than cake or muffin coz we exhausted that set (i.e buyers buying cake or muffin or both), hence these 30 guys are left to buy only breads (this is somewhat tacit)

[gabriel]

Hi Gabriel,
50 + 40 - 20 =70

this figure represents the nos of buyers who buy either cake or muffin or both

so 100 -70 =30 would be the nos of buyers left to buy anything from except cake or muffin, which is bread only. Ya if there were a couple of more poducts then, then we could not have made such a statement.

now since there are only 3 products & these 30 guys are buyers who buy something other than cake or muffin coz we exhausted that set (i.e buyers buying cake or muffin or both), hence these 30 guys are left to buy only breads (this is somewhat tacit)

.. thats not true ..

Let me explain .. the question says that there are 50 people who buy cake mix .. in this 50 people is included 4 groups the people who buy only cake mix, people who buy cake mix and muffin mix, people who buy cake mix and bread mix and people who buy cake mix , muffin mix and bread mix...

now, when you subtract 20( the number of people who buy cake mix and muffin mix) from the number of people who buy cake mix, you are still left with people who cake mix and people who buy bread mix and cake mix ..

I dont know what the answer is yet ( havent solved it yet ) but just wanted to tell you that the logic ur using to get to the figure seems flawed to me ..


PS: - yeah just checked then answer seems to be 3/10 .. but still somehow feel that the reasoning is a little flawed ..

[samirpandeyit62]

Thanks Gabriel,
I did think about this after reading your comment carefully (after posting my reply), but the point is that although we can find out the nos of people who did not buy cake or muffin or both, but if you assume that some people who brought cake also brought bread, & some people who brought muffin also brought bread & some of the people brought all three, then with that we cannot solve the problem, as these nos can be anything, So if this was a critical reasoning question then ofcourse it would not have SUFF or we are asked to find the lowest or highest probabilty, but since it needs to be solved so I solved it to get an ans among the ans choices, anyway thanks for highlighting this, It would be really helpful for lot of people to learn that set theory requires a lot of thinking.

I have one more request I posted in my earlier comment pls provide ur valuable comments on this

How will you interpret the following two statements in context of a set theory problem

stmt 1 : The nos of buyers who buy cake (C) or muffin (M) is 50

Acoording to me this should be C Or M

i.e people who buy cake or muffin or both

i.e C + M -C and M (people who buy both)

stmt 2: The nos of buyers who buy either cake or muffin is 50

does this mean people who buy cake or muffin but not both.
i.e. C + M - 2(C and M) (people who buy both)

[gabriel]

...

How will you interpret the following two statements in context of a set theory problem

stmt 1 : The nos of buyers who buy cake (C) or muffin (M) is 50

Acoording to me this should be C Or M

i.e people who buy cake or muffin or both

i.e C + M -C and M (people who buy both)

stmt 2: The nos of buyers who buy either cake or muffin is 50

does this mean people who buy cake or muffin but not both.
i.e. C + M - 2(C and M) (people who buy both)

Hi Samir..

The interpretation seems fine to me .. I mean that is how i would interpret it too ....

when it is said cake or muffin .. then it would mean cake, muffin or both .. so that would indeed mean C+M-(C & M)

and when someone says either cake or muffin ... then we cannot consider the case of both cake and muffin .. that would mean C+M-2(C & M) ..

Regards ..

[samirpandeyit62]

Thanks very much Gabriel.