PS Overlapping sets

[sud21]

200 people responded to a survey that asked them to rank three different brands of soap. The percentage of respondents that ranked each brand 1st, 2nd, and 3rd are listed above.
If no respondents rated the soaps in the order Y, Z, X, how many respondents rated the soap in the following order: X, Y, Z?

100

60

50

30

25

[theCEO]

200 people responded to a survey that asked them to rank three different brands of soap. The percentage of respondents that ranked each brand 1st, 2nd, and 3rd are listed above.
If no respondents rated the soaps in the order Y, Z, X, how many respondents rated the soap in the following order: X, Y, Z?

100

60

50

30

25

There a 6 ways in which we can order x, y, z where the first letter is the first position, etc
x y z
x z y
y x z
y z x
z x y
z y x

We are told that y,z,x has no respondent so if we label each available option, we have
a = x y z
b = x z y
c = y x z
d = z x y
e = z y x

we are told that 15% of the respondent choose x as the last choice,
therefore e = 15% x 200 = 30

we are told that 40% of the respondent choose y as the second choice
therefore a + e = 40% x 200 = 80
therefore a = 80 - e = 80 - 30 = 50

therefore 50 people choose x,y,z

ans = c

[Matt@VeritasPrep]

Since nobody ranked the soaps Y > Z > X, we know that EVERYONE who thought Y was best must have given the order Y > X > Z. (If you rank Y #1, these are the only two arrangements.)

So Y > X > Z = 60 people.

Using similar logic, since nobody ranked the soaps Y > Z > X, we know that EVERYONE who thought Z was second best must have given the order X > Z > Y.

So X > Z > Y = 30 people.

Since there are only two ways to rate X best (either X > Z > Y or X > Y > Z), these two arrangements must sum to 80, the number of people who rated X best. We have 30 already from X > Z > Y, so the number who gave X > Y > Z must be 50.

[sud21]

Hi, theceo,

how are you getting 15%? Should not it be 20% (1/5)?

[theCEO]

Hi, theceo,

how are you getting 15%? Should not it be 20% (1/5)?

Hi sud21,

In order to find the percentage of respondent that rank z > y > x, we need to find the graph that addresses x when x is the 3rd choice. Looking at graph #3, we see that 15% of respondent choose x as the 3rd choice.

15% x 200 respondent =30 respondent choose x as the last choice.

Let me know if this help.

How did you get 20%?

[sud21]

I got that. I got 20% by picking one scenario where x is last out of the 5 scenarios.

[theCEO]

I got that. I got 20% by picking one scenario where x is last out of the 5 scenarios.

Understood. If the graphs were not presented then there will be a 20% chance that X is the last choice.