Understanding Conditional Probability (Math is Fun)

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I have trouble understanding a simple concept from Math is Fun.

STATEMENT: 70% of your friends like Chocolate, and 35% like Chocolate AND like Strawberry. What percent of those who like Chocolate also like Strawberry?

SOLUTION: P(Strawberry|Chocolate) = P(Chocolate and Strawberry) / P(Chocolate)

0.35 / 0.7 = 50% . Hence 50% of your friends who like Chocolate also like Strawberry

WHAT I CAN'T UNDERSTAND: What is the difference between the statements that "35% of friends like both Chocolate and Strawberry" and "friends who like Chocolate also like Strawberry". Just a play of words, practically they seem exact same to me. Am I missing something ?

probability

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asked 3 hours ago

Arnuld

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up vote
5
down vote

favorite

I have trouble understanding a simple concept from Math is Fun.

STATEMENT: 70% of your friends like Chocolate, and 35% like Chocolate AND like Strawberry. What percent of those who like Chocolate also like Strawberry?

SOLUTION: P(Strawberry|Chocolate) = P(Chocolate and Strawberry) / P(Chocolate)

0.35 / 0.7 = 50% . Hence 50% of your friends who like Chocolate also like Strawberry

WHAT I CAN'T UNDERSTAND: What is the difference between the statements that "35% of friends like both Chocolate and Strawberry" and "friends who like Chocolate also like Strawberry". Just a play of words, practically they seem exact same to me. Am I missing something ?

probability

share|cite|improve this question

asked 3 hours ago

Arnuld

11211

add a comment | 

up vote
5
down vote

favorite

up vote
5
down vote

favorite

I have trouble understanding a simple concept from Math is Fun.

STATEMENT: 70% of your friends like Chocolate, and 35% like Chocolate AND like Strawberry. What percent of those who like Chocolate also like Strawberry?

SOLUTION: P(Strawberry|Chocolate) = P(Chocolate and Strawberry) / P(Chocolate)

0.35 / 0.7 = 50% . Hence 50% of your friends who like Chocolate also like Strawberry

WHAT I CAN'T UNDERSTAND: What is the difference between the statements that "35% of friends like both Chocolate and Strawberry" and "friends who like Chocolate also like Strawberry". Just a play of words, practically they seem exact same to me. Am I missing something ?

probability

share|cite|improve this question

asked 3 hours ago

Arnuld

11211

I have trouble understanding a simple concept from Math is Fun.

STATEMENT: 70% of your friends like Chocolate, and 35% like Chocolate AND like Strawberry. What percent of those who like Chocolate also like Strawberry?

SOLUTION: P(Strawberry|Chocolate) = P(Chocolate and Strawberry) / P(Chocolate)

0.35 / 0.7 = 50% . Hence 50% of your friends who like Chocolate also like Strawberry

WHAT I CAN'T UNDERSTAND: What is the difference between the statements that "35% of friends like both Chocolate and Strawberry" and "friends who like Chocolate also like Strawberry". Just a play of words, practically they seem exact same to me. Am I missing something ?

probability

probability

share|cite|improve this question

asked 3 hours ago

Arnuld

11211

share|cite|improve this question

asked 3 hours ago

Arnuld

11211

share|cite|improve this question

share|cite|improve this question

asked 3 hours ago

Arnuld

11211

asked 3 hours ago

Arnuld

11211

asked 3 hours ago

Arnuld

11211

11211

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3 Answers
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The small difference in the words implies a difference in the denominator in the calculation, while the numerator stays the same.

Suppose you in fact have $20$ friends of whom $14$ like Chocolate, and of these $7$ like Chocolate and Strawberry.

• The proportion of all your friends who like both is $dfrac720=35%$
  


• The proportion of your Chocolate-liking friends who also like Strawberry is $dfrac714=50%$
  

This second calculation is equivalent to $frac7/2014/20 = frac35%70%$, and you would get the same percentage result if you in fact have $100$ or $2000$ friends

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answered 2 hours ago

Henry

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The difference is in what comes before.

For chocolate and strawberry the question is explicitly looking at everyone, and then within them, who likes both.

The second bit; those who like chocolate also like strawberry, means that you are looking at a smaller group to begin with. You look at those who like chocolate, and within that smaller group find those who like both.

Obviously those who like chocolate who also like strawberry are in the group of those who like chocolate and strawberry. So you're right in saying that they mean the same thing - to an extent. It just changes how you view the "population".

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answered 2 hours ago

MRobinson

1,04518

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Let me know if that doesn't make any sense and I'll try and reword
  – MRobinson
  2 hours ago
  

  
  
  
  

add a comment | 

up vote
0
down vote

I think there is a confusion about $measures$:

we do not discuss the $share$ or number of people in the population, we discuss another measure, called $probability$, which sometimes can be viewed as a share or proportion, but not necessarily. In your case $70 %$ means 'pick a person from the population, what's the chance he likes chocolate' and $50 %$ means 'pick a person from the population, what's the chance he likes chocolate'.

$35 %$ means 'pick a person from the group of people that like chocolate (subset of the population), what's the chance he likes the strawberry'. So you restrict the $selection$ to a subset of the population. Therefore it is not an absolute number that matters, but a 'chance to observe an effect', not by examining the whole population, but a subset thereof.

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answered 22 mins ago

Alex

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3 Answers
3

active

oldest

votes

3 Answers
3

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
3
down vote

The small difference in the words implies a difference in the denominator in the calculation, while the numerator stays the same.

Suppose you in fact have $20$ friends of whom $14$ like Chocolate, and of these $7$ like Chocolate and Strawberry.

• The proportion of all your friends who like both is $dfrac720=35%$
  


• The proportion of your Chocolate-liking friends who also like Strawberry is $dfrac714=50%$
  

This second calculation is equivalent to $frac7/2014/20 = frac35%70%$, and you would get the same percentage result if you in fact have $100$ or $2000$ friends

share|cite|improve this answer

answered 2 hours ago

Henry

94.6k473151

add a comment | 

up vote
3
down vote

The small difference in the words implies a difference in the denominator in the calculation, while the numerator stays the same.

Suppose you in fact have $20$ friends of whom $14$ like Chocolate, and of these $7$ like Chocolate and Strawberry.

• The proportion of all your friends who like both is $dfrac720=35%$
  


• The proportion of your Chocolate-liking friends who also like Strawberry is $dfrac714=50%$
  

This second calculation is equivalent to $frac7/2014/20 = frac35%70%$, and you would get the same percentage result if you in fact have $100$ or $2000$ friends

share|cite|improve this answer

answered 2 hours ago

Henry

94.6k473151

add a comment | 

up vote
3
down vote

up vote
3
down vote

The small difference in the words implies a difference in the denominator in the calculation, while the numerator stays the same.

Suppose you in fact have $20$ friends of whom $14$ like Chocolate, and of these $7$ like Chocolate and Strawberry.

• The proportion of all your friends who like both is $dfrac720=35%$
  


• The proportion of your Chocolate-liking friends who also like Strawberry is $dfrac714=50%$
  

This second calculation is equivalent to $frac7/2014/20 = frac35%70%$, and you would get the same percentage result if you in fact have $100$ or $2000$ friends

share|cite|improve this answer

answered 2 hours ago

Henry

94.6k473151

The small difference in the words implies a difference in the denominator in the calculation, while the numerator stays the same.

Suppose you in fact have $20$ friends of whom $14$ like Chocolate, and of these $7$ like Chocolate and Strawberry.

• The proportion of all your friends who like both is $dfrac720=35%$
  


• The proportion of your Chocolate-liking friends who also like Strawberry is $dfrac714=50%$
  

This second calculation is equivalent to $frac7/2014/20 = frac35%70%$, and you would get the same percentage result if you in fact have $100$ or $2000$ friends

share|cite|improve this answer

answered 2 hours ago

Henry

94.6k473151

share|cite|improve this answer

share|cite|improve this answer

answered 2 hours ago

Henry

94.6k473151

answered 2 hours ago

Henry

94.6k473151

answered 2 hours ago

Henry

94.6k473151

94.6k473151

add a comment | 

add a comment | 

up vote
0
down vote

The difference is in what comes before.

For chocolate and strawberry the question is explicitly looking at everyone, and then within them, who likes both.

The second bit; those who like chocolate also like strawberry, means that you are looking at a smaller group to begin with. You look at those who like chocolate, and within that smaller group find those who like both.

Obviously those who like chocolate who also like strawberry are in the group of those who like chocolate and strawberry. So you're right in saying that they mean the same thing - to an extent. It just changes how you view the "population".

share|cite|improve this answer

answered 2 hours ago

MRobinson

1,04518

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Let me know if that doesn't make any sense and I'll try and reword
  – MRobinson
  2 hours ago
  

  
  
  
  

add a comment | 

up vote
0
down vote

The difference is in what comes before.

For chocolate and strawberry the question is explicitly looking at everyone, and then within them, who likes both.

The second bit; those who like chocolate also like strawberry, means that you are looking at a smaller group to begin with. You look at those who like chocolate, and within that smaller group find those who like both.

Obviously those who like chocolate who also like strawberry are in the group of those who like chocolate and strawberry. So you're right in saying that they mean the same thing - to an extent. It just changes how you view the "population".

share|cite|improve this answer

answered 2 hours ago

MRobinson

1,04518

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Let me know if that doesn't make any sense and I'll try and reword
  – MRobinson
  2 hours ago
  

  
  
  
  

add a comment | 

up vote
0
down vote

up vote
0
down vote

The difference is in what comes before.

For chocolate and strawberry the question is explicitly looking at everyone, and then within them, who likes both.

The second bit; those who like chocolate also like strawberry, means that you are looking at a smaller group to begin with. You look at those who like chocolate, and within that smaller group find those who like both.

Obviously those who like chocolate who also like strawberry are in the group of those who like chocolate and strawberry. So you're right in saying that they mean the same thing - to an extent. It just changes how you view the "population".

share|cite|improve this answer

answered 2 hours ago

MRobinson

1,04518

The difference is in what comes before.

For chocolate and strawberry the question is explicitly looking at everyone, and then within them, who likes both.

The second bit; those who like chocolate also like strawberry, means that you are looking at a smaller group to begin with. You look at those who like chocolate, and within that smaller group find those who like both.

Obviously those who like chocolate who also like strawberry are in the group of those who like chocolate and strawberry. So you're right in saying that they mean the same thing - to an extent. It just changes how you view the "population".

share|cite|improve this answer

answered 2 hours ago

MRobinson

1,04518

share|cite|improve this answer

share|cite|improve this answer

answered 2 hours ago

MRobinson

1,04518

answered 2 hours ago

MRobinson

1,04518

answered 2 hours ago

MRobinson

1,04518

1,04518

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Let me know if that doesn't make any sense and I'll try and reword
  – MRobinson
  2 hours ago
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Let me know if that doesn't make any sense and I'll try and reword
  – MRobinson
  2 hours ago
  

  
  
  
  

Let me know if that doesn't make any sense and I'll try and reword
– MRobinson
2 hours ago

Let me know if that doesn't make any sense and I'll try and reword
– MRobinson
2 hours ago

add a comment | 

up vote
0
down vote

I think there is a confusion about $measures$:

we do not discuss the $share$ or number of people in the population, we discuss another measure, called $probability$, which sometimes can be viewed as a share or proportion, but not necessarily. In your case $70 %$ means 'pick a person from the population, what's the chance he likes chocolate' and $50 %$ means 'pick a person from the population, what's the chance he likes chocolate'.

$35 %$ means 'pick a person from the group of people that like chocolate (subset of the population), what's the chance he likes the strawberry'. So you restrict the $selection$ to a subset of the population. Therefore it is not an absolute number that matters, but a 'chance to observe an effect', not by examining the whole population, but a subset thereof.

share|cite|improve this answer

answered 22 mins ago

Alex

14k42032

add a comment | 

up vote
0
down vote

I think there is a confusion about $measures$:

we do not discuss the $share$ or number of people in the population, we discuss another measure, called $probability$, which sometimes can be viewed as a share or proportion, but not necessarily. In your case $70 %$ means 'pick a person from the population, what's the chance he likes chocolate' and $50 %$ means 'pick a person from the population, what's the chance he likes chocolate'.

$35 %$ means 'pick a person from the group of people that like chocolate (subset of the population), what's the chance he likes the strawberry'. So you restrict the $selection$ to a subset of the population. Therefore it is not an absolute number that matters, but a 'chance to observe an effect', not by examining the whole population, but a subset thereof.

share|cite|improve this answer

answered 22 mins ago

Alex

14k42032

add a comment | 

up vote
0
down vote

up vote
0
down vote

I think there is a confusion about $measures$:

we do not discuss the $share$ or number of people in the population, we discuss another measure, called $probability$, which sometimes can be viewed as a share or proportion, but not necessarily. In your case $70 %$ means 'pick a person from the population, what's the chance he likes chocolate' and $50 %$ means 'pick a person from the population, what's the chance he likes chocolate'.

$35 %$ means 'pick a person from the group of people that like chocolate (subset of the population), what's the chance he likes the strawberry'. So you restrict the $selection$ to a subset of the population. Therefore it is not an absolute number that matters, but a 'chance to observe an effect', not by examining the whole population, but a subset thereof.

share|cite|improve this answer

answered 22 mins ago

Alex

14k42032

I think there is a confusion about $measures$:

we do not discuss the $share$ or number of people in the population, we discuss another measure, called $probability$, which sometimes can be viewed as a share or proportion, but not necessarily. In your case $70 %$ means 'pick a person from the population, what's the chance he likes chocolate' and $50 %$ means 'pick a person from the population, what's the chance he likes chocolate'.

$35 %$ means 'pick a person from the group of people that like chocolate (subset of the population), what's the chance he likes the strawberry'. So you restrict the $selection$ to a subset of the population. Therefore it is not an absolute number that matters, but a 'chance to observe an effect', not by examining the whole population, but a subset thereof.

share|cite|improve this answer

answered 22 mins ago

Alex

14k42032

share|cite|improve this answer

share|cite|improve this answer

answered 22 mins ago

Alex

14k42032

answered 22 mins ago

Alex

14k42032

answered 22 mins ago

Alex

14k42032

14k42032

add a comment | 

add a comment | 

 

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