I had a discussion with a peer about the probability of passing an exam given two different grading schemes and I'm not sure I believe what my peer stated.
They stated that given an exam with 10 questions where a passing grade is 5 correct questions and an exam with 6 questions where a passing grade is 3 questions, it would be better to take the exam with 10 questions as there is an increased probability of passing. I didn't buy the argument as it seems that the exams are equivalent, i.e., you need a 50% to pass either. However my peer was adamant about the their point. Can anyone clarify this?
Best Answer
The rightness or wrongness of your peer's statement depends on the probability of success of answering each question correctly.
If you assume that the test is a set of $2N$ true/false questions, with N correct answers required to pass, where your probability of answering any question is $p$, then the probability $P$ of passing the test is such that:
for $p<0.5$, $P$ falls monotonically with increasing N and in the limit of $N {\rightarrow} {\infty}$, $P {\rightarrow} 0$, so it will always be preferential to choose the test with the least number of questions.
for $p=0.5$ the probability of passing still falls with increasing N (but now asymptotes to 0.5), $N {\rightarrow} {\infty}$, $P {\rightarrow} 0.5$, so still choose the test with the least number of questions.
for $0.5<p<2/3$ the probability of passing initially falls with increasing N, but then increases with larger N and in the limit $N {\rightarrow} {\infty}$, $P {\rightarrow} 1.0$, so your choice would depend on the maximum number of questions. For example, if $p=0.51$ then sitting a test with $N\simeq570$ questions is marginally better than sitting a test with $N=2$ questions.
for $p>2/3$ the probability of passing increases monotonically with increasing N, and in the limit $N {\rightarrow} {\infty}$, $P {\rightarrow} 1.0$, so you should always choose the test with the most questions.
In your example, choosing either a 6 question or a 10 question test, your probability of success will be approximately equal if $p\simeq0.564$ (in that case $P\simeq0.7674$), it would be better to do the 6 question test if $p<0.564$, but you should choose the 10 question test if $p>0.564$.