probability issues

Kashmirnutshell

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Manufactuer is testing 4 brands of soda in a blind taste test. The participants know brands being tested but do not know which is which. What is the probability that a participant will identify all 4 brands correctly by guessing

A. .4 B. .04 C. .1 D .01
 

MarkFL

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Hello, and welcome to FMH! :)

How many ways are there to arrange 4 distinct objects in 4 locations (such as a lineup)?
 

Kashmirnutshell

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Thank you for replying, but thats my trouble. Im trying to learn the method to figure that out. Seems like You multiply 4 by 4 but thats not right
 

MarkFL

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Thank you for replying, but thats my trouble. Im trying to learn the method to figure that out. Seems like You multiply 4 by 4 but thats not right
Suppose you are going to place the 4 drinks in a line, and you have the spots for them in front of you. You are going to put one of the drinks in the first spot. How many drinks do you have from which to choose?

How about for the second spot, and so on?
 

Jomo

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In my opinion NONE of those choices are correct.
In how many different ways can participant guess all 4 correctly and how many different ways can a participant rank the 4 sodas
 

Kashmirnutshell

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3 and then 2 and then 1
 

Jomo

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Thank you for replying, but thats my trouble. Im trying to learn the method to figure that out. Seems like You multiply 4 by 4 but thats not right
Call the sodas A, B, C and D. How many ways can you write down just those four letters. I will start you off
ABCD
ABDC
ACBD
ABDC
keep going....
Now once you figure out all the possibly outcomes, then you need to figure out how many of those outcomes will be correct.
 

Kashmirnutshell

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In my opinion NONE of those choices are correct.
In how many different ways can participant guess all 4 correctly and how many different ways can a participant rank the 4 sodas
Yes your right there is an approx sign in front of the choices. Im stuck on the method you use to determine how many possibilities there are
 

MarkFL

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3 and then 2 and then 1
Yes, 4 choices for the first spot, 3 for the second, 2 for the third, and 1 for the fourth. Using the fundamental counting principle, then how many ways can you order the 4 drinks?
 

Kashmirnutshell

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Call the sodas A, B, C and D. How many ways can you write down just those four letters. I will start you off
ABCD
ABDC
ACBD
ABDC
keep going....
ADBC
ACDB
ADCB
BACD
BCAD
BDAC
BDCA
BCDA
Seems like this can go forever. There isnt a method to determine that with out writing all that out? Guys like me have to write all that but you guys can do this stuff in your head
 

Kashmirnutshell

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Yes, 4 choices for the first spot, 3 for the second, 2 for the third, and 1 for the fourth. Using the fundamental counting principle, then how many ways can you order the 4 drinks?
10?
 

Kashmirnutshell

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ksdhart2

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Well, in your previous brute force attempt, you listed more than 10 possibilities, so we know for sure that can't be the answer. This suggests that simply adding the number of possibilities is not correct. But let's investigate why that might be, and see if we can't stumble upon what is the correct answer. Suppose for a moment there were only two slots and we could choose from four sodas for the first slot and choose from three for the second slot. Right away we can begin listing the possibilities for the first slot (again, labeling the sodas A, B, C, and D):
  • A[?]
  • B[?]
  • C[?]
  • D[?]
Here [?] denotes a "wildcard" - that particular slot can be any of the three other sodas. We know that the second slot has three possibilities, so we can fill in the missing information:
  • A[?]
    • AB
    • AC
    • AD
  • B[?]
    • BA
    • BC
    • BD
  • C[?]
    • CA
    • CB
    • CD
  • D[?]
    • DA
    • DB
    • DC
In other words, for each of the four sodas we can put in the first slot, we have three choices for the second slot. The words I've bolded are the very important bit, as thinking about what those words mean with reference to word problems should give you an inclination as to why it makes intuitive sense to multiply rather than add.

Edit: To appease the grammar gods, I've inserted a few words to make my meaning extremely explicit. These inserted words are underlined.
 
Last edited:

Jomo

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ADBC
ACDB
ADCB
BACD
BCAD
BDAC
BDCA
BCDA
Seems like this can go forever. There isnt a method to determine that with out writing all that out? Guys like me have to write all that but you guys can do this stuff in your head
Yes!, Guys like you have to write all that but we guys can do this stuff in your head. The way we learned the short cuts was by seeing what these lists look like and then try to figure out shortcut.

The list will not go on forever. Now please complete the list
 

Jomo

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Well, in your previous brute force attempt, you listed more than 10 possibilities, so we know for sure that can't be the answer. This suggests that simply adding the number of possibilities is not correct. But let's investigate why that might be, and see if we can't stumble upon what is the correct answer. Suppose for a moment there were only two slots and we could choose four sodas for the first slot and three for the second slot. Right away we can begin listing the possibilities for the first slot (again, labeling the sodas A, B, C, and D):
  • A[?]
  • B[?]
  • C[?]
  • D[?]
Here [?] denotes a "wildcard" - that particular slot can be anything. We know that the second slot has three possibilities, so we can fill in the missing information:
  • A[?]
    • AB
    • AC
    • AD
  • B[?]
    • BA
    • BC
    • BD
  • C[?]
    • CA
    • CB
    • CD
  • D[?]
    • DA
    • DB
    • DC
In other words, for each of the four sodas we put in the first slot, we have three choices for the second slot. The words I've bolded are the very important bit, as thinking about what those words mean with reference to word problems should give you an inclination as to why it makes intuitive sense to multiply rather than add.
No, you are NOT putting four sodas in the first slot. You are putting ONE SODA from a possible four into the 1st slot.
 

MarkFL

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Lol thank you for your help. As You can see, i need it. 24?
Yes, there are \(4!=24\) different ways to order the drinks. In general, there are \(n!\) ways to order \(n\) objects. So, can you now answer the question?
 

ksdhart2

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No, you are NOT putting four sodas in the first slot. You are putting ONE SODA from a possible four into the 1st slot.
Fine. :rolleyes: If it will make you happy, I'll insert some words to "fix" my minor grammatical errors that anyone with any bit of proficiency in English would gloss over because my meaning is fairly clear.
 

MarkFL

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No, there are 24 ways to order the 4 drinks, but only one of those ways will correspond to the particular sequence the taste tester will guess.
 
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