# Understanding Conditional Probability (Math is Fun)

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 ?

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 $$\dfrac{7}{20}=35\%$$
• The proportion of your Chocolate-liking friends who also like Strawberry is $$\dfrac{7}{14}=50\%$$

This second calculation is equivalent to $$\frac{7/20}{14/20} = \frac{35\%}{70\%}$$, and you would get the same percentage result if you in fact have $$100$$ or $$2000$$ friends

• So, 7 people who like both will always come from 14-group and rest of the 6 people like only Strawberry and not Chocolate. and 7/14 means 50% . They just removed the people who not like Chocolate. So conditional probability means removal of certain group/category of people ? Commented Sep 28, 2018 at 2:52

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".

• Let me know if that doesn't make any sense and I'll try and reword Commented Sep 27, 2018 at 7:17
• You explanation is simple enough. Understanding this in practical terms is bit difficult. Why you used the sentence " they mean the same thing - to an extent". If you look at 100% of the people then those who like both C & S will always come from C group because C group contains all the people who like both C and S. So, it is just that number of total people have been reduced from 100 to 70 because all the people who do not like C have been removed from the total. Am I correct ? Commented Sep 28, 2018 at 2:46
• @Arnuld Yes, exactly. It can be important to look past the language in questions like this and think about what they actually want Commented Sep 28, 2018 at 7:15

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 $$35 \%$$ means 'pick a person from the population, what's the chance he likes strawberry'.

$$35 \%$$ 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', of 'likelihood to observing the effect' not by examining the whole population, but a subset thereof.

• I am sorry but I find it very hard to relate conditional probability to practical life situation . People who like Strawberry and Chocolate are exactly the same ones who like Chocolate and Strawberry. Does the order of priority matter ? I like both but I find Chocolate tastier than Strawberry. Is this what $P(Strawberry | Chocolate)$ means ? In this case it is no longer a game of numbers but emotions/taste came into the picture. Commented Sep 27, 2018 at 17:05
• @Arnuld Preference between the two doesn't matter. $P(Strawberry|Chocolate)$ means probability they like strawberry given they like chocolate. So you look at all the people who like chocolate, and see what proportion of those like strawberry. Commented Sep 28, 2018 at 7:17
• @MRobinson I think the OP has already heard 'like strawberry given they like chocolate'. What I think he doesn't understand that we are not talking about ABSOLUTE numbers, but the $likelihood$ to observe a strawberry lover whether we consider the whole population vs subset (choc lovers). Does this make sense?
– Alex
Commented Sep 28, 2018 at 11:49
• @Alex Of course it makes sense, that's what I outlined in the second sentence Commented Sep 28, 2018 at 12:58
• @MRobinson sorry the last question was more directed at the OP. I think it's better to look at probability as measure of $likelihood$ or $opinion$ rather than share. proportion or absolute number. Our $opinion$ of the preference of strawberry changes when we consider the subset of the population rather than the whole population.
– Alex
Commented Sep 28, 2018 at 14:52