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Difficulty:
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The table above shows the number of residents in each of two age groups who support the use of each type of funding for a city initiative. What is the probability that a person randomly selected from among the 250 residents polled is younger than 40, or supports a type of funding that includes a tax, or both?
A. 1/5
B. 8/25
C. 12/25
D. 3/5
E. 4/5
PS43661.01
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2019-09-22_0105.png [ 8.49 KiB | Viewed 51053 times ]
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gmatt1476
Total Residents = 250
Let Younger than 40 condition be A and supports a type of funding that includes a tax condition be B
So we need to find \(\frac{P(A ∪ B)}{P(Total)}\) --- (1)
\(P(A ∪ B) = P(A) + P(B) - P(A ∩ B)\) --- (2)
Now,
P(A) = P(Younger than 40 condition) = 20 + 30 + 30 = 80 --- (3)
P(B) = P (supports a type of funding that includes a tax condition) = 20 + 30 +60 + 10 = 120 --- (4)
P(A ∩ B) = P (Both younger than 40 and Including Tax) = 20 + 30 --- (5)
Using (2), (3), (4), (5)
\(P(A ∪ B) = 80 + 120 - 50\)
\(P(A ∪ B) = 150\) --- (6)
Using (1) and (6),
\(\frac{P(A ∪ B)}{P(Total)}\) = \(\frac{150}{250}\) = \(\frac{3}{5}\)
Hence D
giovannisumano
"both" means having both characteristics i.e. is less than 40 and supports tax.
We are given the number of residents who are "< 40" and "40 or more" and those who "support tax", "support fees" and "support both tax and fees". So the figures are very easy to work with.
e.g. how many residents less than 40 support only tax? 20
How many residents < 40 support both tax and fees? 30
etc
We need the number of residents who are either "<40" or "support includes tax".
We know that overall 20 + 30 + 30 = 80 residents are "< 40".
We know that of the residents "40 or more", 10 + 60 = 70 support taxes.
We need "a person that is younger than 40, or supports a type of funding that includes a tax, or both?"
80 people are < 40 and this includes those who are < 40 and support tax. So "both" is taken care of in this 80. Now we just add to it the number of people 40 or more who support tax i.e. 70.
Fav cases = 80 + 70 = 150
Total = 250
P = 150/250 = 3/5
Answer (D)
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General Discussion
Archit3110
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Total residents given ; 250
18-39 age total 80
and support tax or both 80+10+60 ; 150
hence P ;
probability that a person randomly selected from among the 250 residents polled is younger than 40, or supports a type of funding that includes a tax, or both
150/250; 3/5
IMO D
18-39 age total 80
and support tax or both 80+10+60 ; 150
hence P ;
probability that a person randomly selected from among the 250 residents polled is younger than 40, or supports a type of funding that includes a tax, or both
150/250; 3/5
IMO D
gmatt1476
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Schools: LBS '21 INSEAD Jan '19 IIMB EPGP"20 Yale '22 IMD '22 Kellogg '24 NUS Wharton '24 Said '23 (II)
GMAT 1: 740 Q49 V41
Posts: 22
Archit3110
Hi,
Thanks for answering. Please, can you help me with this?
While considering the group "supports a type of funding that includes a tax, or both", you have considered a group of people >40 and that counts to 60+10, but why hasn't the group below <40 is considered that "supports a type of funding that includes a tax, or both" which would have added 50 more?
Thanks,
Akshay
Archit3110
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Akshay_Naik
question has asked ; What is the probability that a person randomly selected from among the 250 residents polled is younger than 40, or supports a type of funding that includes a tax, or both
younger than 40 ; 80 which includes tax and both as well
and other group>40 ; tax and both ; 10+60 ; 70
hope this helps
question has asked ; What is the probability that a person randomly selected from among the 250 residents polled is younger than 40, or supports a type of funding that includes a tax, or both
younger than 40 ; 80 which includes tax and both as well
and other group>40 ; tax and both ; 10+60 ; 70
hope this helps
Akshay_Naik
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gmatt1476
Probability is the number of residents described in conditions divided by 250.
It has a trap answer of 4/5 i.e E. Please make sure you count the resident only once as they overlap in the conditions specified.
Condition1: Residents Less than 40, add all in the first row= 80 residents
Condition2: Residents that pay Tax, 10 This includes two groups; make sure to consider just residents above 40 since the other residents who pay tax are included in condition1 already. It does not make sense to add them again.
Condition3: Residents above 40 and support tax and fees- 60
P=60+10+80/250
Ans: C
Schools: ISB '21 (A)
Posts: 63
gmatt1476
Hello chetan2u,
Could you please help me out here. Why are we worrying about overlapping here? Normally whenever there is an 'OR' in a probability question we just add the asked entities. Why do we have to care about the overlapping here?
Thanks in advance.
RK007
Hi
The reason is that when we are adding ways in which a person is below 40 and ways in which a person is tax, there are person common to both groups, that is those under 40 and paying tax. These people have been added twice , once in each group, and so we have to find overlap and subtract once to remove duplicacy
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Another approach would be to take the inverse route and cancel out all the relevant groups that don't qualify for the stated stem criteria.
We want to find those that are not described in the stem, so here goes:
1) We don't want anyone below age 40 (cross out first row)
Age | Tax Only | Tax & Fees | Fees Only
18-39| X X X
>=40| 10 60 100
2) We don't want anyone with 'Tax Only' or 'Tax & Fees' (cross downwards, anything under 'Tax Only' or 'Tax and Fees')
Age | Tax Only | Tax & Fees | Fees Only
18-39| X X X
>=40| X X 100
3) See that we're left with the inverse situation - 100 people that are not described by the stem.
Therefore probability = 1 - P(Not Described) = 1 - (100/250) = P(Described) = 150/250 = 3/5.
Answer: (D)
We want to find those that are not described in the stem, so here goes:
1) We don't want anyone below age 40 (cross out first row)
Age | Tax Only | Tax & Fees | Fees Only
18-39| X X X
>=40| 10 60 100
2) We don't want anyone with 'Tax Only' or 'Tax & Fees' (cross downwards, anything under 'Tax Only' or 'Tax and Fees')
Age | Tax Only | Tax & Fees | Fees Only
18-39| X X X
>=40| X X 100
3) See that we're left with the inverse situation - 100 people that are not described by the stem.
Therefore probability = 1 - P(Not Described) = 1 - (100/250) = P(Described) = 150/250 = 3/5.
Answer: (D)
gmatt1476
In probability, we should know total events, which is 250 here, and the positive outcomes.
Positive outcomes..
1) All \(<40 = x= 20+30+30=80\)
2) \(\geq{40}\) - Those who choose tax or both = y=10+60=70
\(Total = 80+70=150\).
Probability = \(\frac{positive..outcomes}{total..events}= \frac{150}{250}=\frac{3}{5}\)
D
Attachments
2019-09-22_0105.png [ 15.88 KiB | Viewed 38862 times ]
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gmatt1476
For folks having confusion on the double counting, please have a look at the explanation below:-
1. Probability = (Residents younger than 40 + Residents who support tax, independent of their age + Residents who are younger than 40 and support tax)/250
2. Mathematically, we need to ensure that a person is only counted once in the 3 groups. E.g. If person Z is younger than 40 and supports taxes then will count Z only in the first group (younger than 40) and subtract from the second group (support taxes). Vice versa, we can also count in the second group and subtract from first.
3. Residents younger than 40 = Sum of the first row = 20 + 30 + 30 = 80
4. Residents who support tax, independent of their age = Sum of the first two columns = 20 + 30 + 10 + 60 = 120
4.a. But the 20 + 30 (highlighted in red font) is already considered in point 3 and hence to avoid double counting we will subtract it from 4 i.e. 120 - 20 - 30 = 70
5. Residents who are younger than 40 and support tax = Sum of the first two columns from the first row = 20 + 30 = 50
5.a. But again these two groups are already part of point 3 and hence we will not consider them i.e. 50 - 20 - 30 = 0
Plugging the output from points 3, 4.a, and 5.a. into 1, we get
\(\frac{80 + 70 + 0}{250}\) \(=\) \(\frac{150}{250}\) \(=\) \(\frac{3}{5}\)
Ans. D
This question can be solved very quickly. We know the total number of people is 250
1. Take the total number of people that are age < 40. 20+30+30=80 (in red)
2. Take the number of people that support tax (don't count the people we already counted in the < 40 group). 10 + 60 (in green)
80 + 70 = 150
150 / 250 = 3/5
1. Take the total number of people that are age < 40. 20+30+30=80 (in red)
2. Take the number of people that support tax (don't count the people we already counted in the < 40 group). 10 + 60 (in green)
80 + 70 = 150
150 / 250 = 3/5
Attachments
taxproblem.png [ 6.86 KiB | Viewed 33175 times ]
sujoykrdatta
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gmatt1476
Let us understand how there are 250 people:
Age 18-39: we have 20 + 30 + 30 = 80 people
Age ≥ 40: we have 10 + 60 + 100 = 170 people
Thus, total people = 80 + 170 = 250
We need to find the number of people who are either
# younger than 40 ... (a)
OR
# supports a type of funding that includes a tax ... (b)
OR
# (younger than 40) AND (supports a type of funding that includes a tax) ... (c)
Thus, we have:
# younger than 40 = 20 + 30 + 30 = 80 (horizontal: 1st row) ... (a)
OR
# supports a type of funding that includes a tax
= 20+10 = 30 - tax only (vertical: 1st column); 30+60 = 90 - tax & fees (vertical: 2nd column) => 30 + 90 = 120 ... (b)
# But, we have counted "(younger than 40) AND (supports a type of funding that includes a tax) = 20+30 = 50" in both (a) and (b). Thus, we need to remove this once.
Thus, total = 80 + 120 - 50 = 150
Thus, required probability = 150/250 = 3/5
Answer D
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As this is a combination of sets and probability, I did it this way.
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Posts: 1,324
Perfectly stated.
I think most blew this problem up to be a lot harder than it actually is.
The issue becomes not to make a mistake under time pressure and to understand what A OR B implies.
Posted from my mobile device
I think most blew this problem up to be a lot harder than it actually is.
The issue becomes not to make a mistake under time pressure and to understand what A OR B implies.
VeritasKarishma
Posted from my mobile device
gmatt1476
Alternate Solution :
Check one: All the numbers add up to 250. So there are no residents that overlap their decision or chose not to vote.
The only case that we do not want in our probability would be resident>=40 and not supporting tax (i.e. fee only).
As P=1-(~P), the required probability would be 1-(100/250) = 3/5.
Two mutually exclusive magnitudes with sets with intersections.
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adstudy
I initially thought the trap here is that you have to subtract the people who both belong under 40 group and support a type of a tax but in the end it is mentioned "or both"? Doesn't that imply that we need not subtract the common people in the groups? What am I missing? Thanks
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