In February, the Math League students will take the Math
League contest. It
has 40 multiple choice questions with 4 possible answers for
each. I
was wondering how much of a role chance plays in the results.
There's no doubt it plays some part, but how much? If more
students take it, there will be an increase in the likelihood of
high scores. There will also be more low scores, but only the
top 5 count.
A week or so before the contest, we should probably make a
morning announcement about the upcoming tests in case there are
star math students who are interested in taking the test, but
who either were not able to join Math League or had to drop out
for one reason or another. I
did some analysis, and it turns out that chance does not really
play as big a role as I thought it might, but its still an
interesting result.
The
original question I posed to myself was,
How
much would the chance that a student scores greater than 30 just
by random guessing increase if I could get 100 students to take
the test instead of 20?
(This
is sort of like the 100 monkeys typing at a 100 typewriters
reproducing the works of Shakespeare proposition :) )
If
my computations are correct, the answer is - not that much. I
think the probability goes from something like 1 in 10 billion
to only 1 in a billion.
However,
I hope they don't guess for all problems, so maybe it is more
realistic to say that they can get about 10 right without
guessing, and then guess on the remaining 30 (on average). Then
the probability of getting 30 or more right overall might go
from 1.1% for 20 students to 5.5% for 100 students. This is
starting to look a little better.
I
assert that the formula for the probability of 1 student getting
at least numRight right out of numQuestions questions is
SUMMATION i=(0, numQuestions - numRight) {
C(numQuestions,
numRight+i) * (numChoices-1) ^ (numQuestions - numRight - i)
-------------------------------------------------------------------
numChoices ^
numQuestions
}
Where C(a,b) is a choose b = a!/(b!(a-b)!)
and ^ indicates exponentiation.
I
wrote a program to compute this and here are the results:
Probability 0 or more right when taking test is 1.0
(i.e. Its 100% likely that they will get 0 or more
right :))
Probability 1 or more right when taking test is
0.99999
Probability 2 or more right when taking test is
0.99986
Probability 3 or more right when taking test is
0.99898
Probability 4 or more right when taking test is
0.9953 (It very close to certain that they will get
at least 4 by guessing)
Probability 5 or more right when taking test is
0.98396
Probability 6 or more right when taking test is
0.95673
Probability 7 or more right when taking test is
0.90378
Probability 8 or more right when taking test is
0.81805
Probability 9 or more right when taking test is
0.70017
Probability 10 or more right when taking test is
0.56046
Probability 11 or more right when taking test is
0.4161
Probability 12 or more right when taking test is
0.28486
Probability 13 or more right when taking test is
0.17913
Probability 14 or more right when taking test is
0.10323
Probability 15 or more right when taking test is
0.05444 (Only 5% likely that they can get 15 or more right
by guessing)
Probability 16 or more right when taking test is
0.02624
Probability 17 or more right when taking test is
0.01156
Probability 18 or more right when taking test is
0.00465
Probability 19 or more right when taking test is
0.00171
Probability 20 or more right when taking test is
0.00057
Probability 21 or more right when taking test is
0.00017
Probability 22 or more right when taking test is
0.00004864
Probability 23 or more right when taking test is
0.0000123
Probability 24 or more right when taking test is
0.00000283
Probability 25 or more right when taking test is
0.00000058797
Probability 26 or more right when taking test is
0.00000011053
Probability 27 or more right when taking test is
0.00000001871
Probability 28 or more right when taking test is
0.00000000284
Probability 29 or more right when taking test is
3.851E-10
Probability 30 or more right when taking test is
4.631E-11
Probability 31 or more right when taking test is
4.906E-12
Probability 32 or more right when taking test is
4.536E-13
Probability 33 or more right when taking test is
3.618E-14
Probability 34 or more right when taking test is
2.453E-15
Probability 35 or more right when taking test is
1.386E-16
Probability 36 or more right when taking test is
6.35E-18
Probability 37 or more right when taking test is
2.266E-19
Probability 38 or more right when taking test is
5.939E-21
Probability 39 or more right when taking test is
1.323E-22
Probability 40 or more right when taking test is
3.309E-23
Now what happens when there is more than one
student taking the test?
These results compare the results for 20 students
versus 100.
Probability of having at least one student out of
20 get >=20 right is 0.0114
Probability of having at least one student out of
100 get >=20 right is
0.05565
Probability of having at least one student out of
20 get >=30 right is
9.26E-10
Probability of having at least one student out of
100 get >=30 right is
4.63E-9
There is an advantage to having
more students take the contest when drawing from the same
population, but I am sure a much larger factor is the size
of the population (in other words the number of students at
the middle school). If the middle school is large then,
statistically, there should be more students that would
excel at math in general. The other big factor in their
performance is how well we coach them. Hopefully that will
be the biggest factor.