Why can I not simply add fractions statistics?

Question

This is a really elementary question for kids but I have difficulties at the moment to find my mistake. I stepped over the problem in my statistics course at which we use bayes theorem and it appears when I have to give the probability of a certain event:

Say, I ask 100 people, 52 female and 48 male, what kind of superpower they would like to have. 26 male persons picked "flying" and 12 female persons picked flying. What is the probability that someone is male, given that he picked "flying" ? If you use bayes theorem you come very easily to the result of about ~68% :

$ p(male|fly)=\frac{p(fly|male)*p(male)}{p(fly)}$

My question belongs to $p(fly)$ : The probability $p(fly)$ is $38/100$.Because 100 people were asked and in sum 38 of them picked "flying". But I calculated in my first attempt : "26 out of 48 male persons picked flying and 12 out of 52 female persons picked flying $\implies \frac{26}{48}+\frac{12}{52} = \frac{77}{100}$

What is the error of thinking that I did here ? Because I can obviously see that in total 38 of 100 persons picked flying and 77/100 is wrong but I don't understand if 26 of 48 and 12 of 52 picked flying,why adding those does not give me the correct result.

To make the question more clear : Why do we do in this case : $\frac{26+12}{48+52}$ and not $\frac{26}{48}+\frac{12}{52}$,, while having $\frac{26}{48}$ and $\frac{12}{52}$ flying people.

Or another way would be : I have 100 sweets, 48 chocolate taste, 52strawberry taste. I eat 12 strawberry ones and 26 chocolate ones. Why can I not get the total of eaten sweets by (26 out of 48) + (12 out of 52) but by (26+12) out of (100=52+48).

Answer 1

Let's take other numbers: All $52$ females and all $48$ males said they wanted "flying". Your calculation would find that

$$\frac{52}{52} + \frac{48}{48}=2 = 200\%$$

of the asked people wanted to fly.

Your main error is that relative values (like $\frac{26}{48}$) only have an absolute meaning if you attach them to the value they reference (in this case the $48$ males that got asked).

Adding relative values that belong to different reference numbers makes no sense.

Answer 2

If you look at your distribution of the 100 guys (you can do a $2\times 2$ table for that purpose) you see that

• 26 guys are Male and Flying
• 22 guys are Male and NOT Flying
• 12 guys are Female and Flying
• 40 guys are Female and NOT flying,

so, given a guy is Flying ($26+12=38$ possible choices) only 26 are male.

Thus the result is $\frac{26}{38}=\frac{13}{19}\approx 68\%$

Answer 3

You forgot to multiply the probabilities with $\frac{48}{100}$ and $\frac{52}{100}$ respective depending on whether we have a male or female. If you do that, you get the correct result because you add P(male and fly) and P(female and fly)