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The following shows the results of a wine tasting survey of 100 people: 96 like wine A, 93 like wine B, 96 like wine C, 92 like A and B, 91 like B and C, 93 like A and C, 90 like all three wines - Edexcel - A-Level Maths Statistics - Question 5 - 2008 - Paper 1

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The-following-shows-the-results-of-a-wine-tasting-survey-of-100-people:--96-like-wine-A,-93-like-wine-B,-96-like-wine-C,-92-like-A-and-B,-91-like-B-and-C,-93-like-A-and-C,-90-like-all-three-wines-Edexcel-A-Level Maths Statistics-Question 5-2008-Paper 1.png

The following shows the results of a wine tasting survey of 100 people: 96 like wine A, 93 like wine B, 96 like wine C, 92 like A and B, 91 like B and C, 93 like A ... show full transcript

Worked Solution & Example Answer:The following shows the results of a wine tasting survey of 100 people: 96 like wine A, 93 like wine B, 96 like wine C, 92 like A and B, 91 like B and C, 93 like A and C, 90 like all three wines - Edexcel - A-Level Maths Statistics - Question 5 - 2008 - Paper 1

Step 1

Draw a Venn Diagram to represent these data.

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Answer

To represent the survey data in a Venn Diagram:

  • Let A be the set of people who like wine A.
  • Let B be the set of people who like wine B.
  • Let C be the set of people who like wine C.

Fill in the overlaps based on the given data:

  • 90 people like all three wines (A ∩ B ∩ C).

  • For A and B (but not C), there are:

    92 - 90 = 2.

  • For A and C (but not B), there are:

    93 - 90 = 3.

  • For B and C (but not A), there are:

    91 - 90 = 1.

Finally, we calculate those who only like each wine:

  • Only A: 96 - (2 + 3 + 90) = 1.
  • Only B: 93 - (2 + 1 + 90) = 0.
  • Only C: 96 - (3 + 1 + 90) = 2.

The Venn diagram should visually represent these numbers.

Step 2

none of the three wines.

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Answer

The total number of people surveyed is 100. The number of people who like at least one wine can be found from the Venn diagram.

Calculating the total who like at least one wine:

96+93+96929193+90=9996 + 93 + 96 - 92 - 91 - 93 + 90 = 99

Thus, those who like none of the wines:

10099=1100 - 99 = 1

The probability is:

P(none)=1100=0.01P(none) = \frac{1}{100} = 0.01

Step 3

wine A but not wine B.

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Answer

To find the number of people who like wine A but not wine B:

This is simply the number of people who like only A plus those who like A and C (but not B):

P(A¬B)=1+3=4P(A \cap \neg B) = 1 + 3 = 4

The probability is thus:

P(A¬B)=4100=0.04P(A \cap \neg B) = \frac{4}{100} = 0.04

Step 4

any wine in the survey except wine C.

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Answer

We need to find those who like wine A or wine B (but not wine C). From our counts:

  • Only A: 1
  • Only B: 0
  • A and B (not C): 2

Thus:

P(AB¬C)=1+0+2=3P(A \cup B \cap \neg C) = 1 + 0 + 2 = 3

The probability is:

P(anywine¬C)=3100=0.03P(any wine \neg C) = \frac{3}{100} = 0.03

Step 5

exactly two of the three kinds of wine.

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Answer

For exactly two kinds:

  • A and B (not C): 2
  • A and C (not B): 3
  • B and C (not A): 1

Thus:

P(exactlytwo)=2+3+1=6P(exactly two) = 2 + 3 + 1 = 6

The probability will be:

P(exactlytwo)=6100=0.06P(exactly two) = \frac{6}{100} = 0.06

Step 6

find the probability that the person likes wine C.

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Answer

Given that a person from the survey likes wine A:

  • The number who like B and C is 1.
  • Total who like A is 90.

The probability can be found using:

P(CA)=P(CA)P(A) =9090=1P(C|A) = \frac{P(C \cap A)}{P(A)}\ = \frac{90}{90} = 1

Thus, the probability is:

P(CA)=90100=0.90P(C|A) = \frac{90}{100} = 0.90

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