A Kaggle.com question to fight spam is:
A total of 61 students are taking a probability course. 33 are majoring in mathematics, 29 in statistics, 18 in finance, 17 majoring in mathematics and statistics, 9 majoring in mathematics and finance, 10 majoring in statistics and finance, and 7 majoring in all three. What is the number of people not majoring in either mathematics, statistics, or finance?
My approach was to work backwards, by assuming that 7 is a unique number, then subtracting it from the 2-major group figures, then subtracting the sum of 7 + the 2-major group figures from the sum of 33 + 29 + 18 to figure out how many math, stats, and finance student to subtract from the original figure of 61.
I finally got it. My initial approach was correct, but apparently I must've punched in a wrong number somewhere along the line.
My logic was:
Let a = mathematics
Let b = statistics
Let c = finance
- 7 unique students major in a, b, and c
- 3 additional unique students major in b and c (but not a; i.e. 7 of the 10 in the 2nd-to-last group described are the same 7 in a, b, and c)
- 2 additional unique students major in a and c (but not b)
- 10 additional unique students major in a and b (but not c)
- There are
33 + 29 + 18 = 80students described as majoring in a, b, or c, thus this count is of non-distinct records (80 > 61) - Of the 33 majoring in a,
33 - 7 - 10 - 2 = 14are majoring exclusively in a - Of the 29 majoring in b,
29 - 7 - 10 - 3 = 9are majoring exclusively in b - Of the 18 majoring in c,
18 - 7 - 2 - 3 = 6are majoring exclusively in c 14 + 9 + 6 = 29are majoring exclusively in a, b, or c and7 + 3 + 2 + 10 = 22unique students are majoring in some combination thereof- Of the 61 students,
61 - 29 - 22 = 10are not majoring in either mathematics, statistics, or finance
I actually had to do this twice to pass their check, probably because I answer the question above wrong before getting it right.
In the 2nd permutation it went like this:
- 63 total students
- 8 unique students major in math, stats, and finance
- 13 additional unique students major in stats and finance
- 10 additional unique students major in math and finance (but not stats)
7 additional unique students major in math and stats (but not finance) therefore 28 unique multi-majors of math,stats, and/or finance
There are
31 + 32 + 36 = 99students described as majoring in math, stats, or finance, thus this count is of non-distinct records (99 > 63)- Of the 31 majoring in math,
31 - 8 - 10 - 7 = 6are majoring exclusively in math - Of the 32 majoring in stats,
32 - 8 - 13 - 7 = 4are majoring exclusively in stats - Of the 36 majoring in finan,
36 - 8 - 13 - 10 = 5are majoring exclusively in finance 6 + 4 + 5 = 15are majoring exclusively in math, stats, or finance and8 + 13 + 10 + 7 = 38unique students are majoring in some combination thereof- Of the 63 students,
63 - 15 - 38 = 10are not majoring in either mathematics, statistics, or finance
Oy vey! What happened to the anti-spam questions where you click all the pictures containing pizza?
On my first pass I had 1 number off in 1 equation and got an answer of 12. Kaggle.com actually did accept this as the correct answer for some reason, but @N.F. Taussig has convinced me that the answer is 10 not 12 in his comments and solutions. Thanks!
The Inclusion-Exclusion Principle states that
$$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$$
where $|S|$ denotes the number of elements in set $S$. Thus, the number of students in the class majoring in at least one of the three subjects is
\begin{align*} |A \cup B \cup C| & = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|\\ & = 33 + 29 + 18 - 17 - 9 - 10 + 7\\ & = 51 \end{align*} Since there are a total of $61$ students in the class, there are $10$ students taking the probability class who are not majoring in at least one of the subjects mathematics, statistics, and finance.