Does the probability of guessing a perfect multiple choice test score increase by taking the test multiple times?

Question

Me and my brother are currently trying to solve a debate.

The question is: Does the probability of a student guessing a perfect score on a multiple choice test go up if the student takes the test multiple times over a set period of time.

This started by us discussing the probability of guessing every question on the SAT, resulting in a perfect score. He theorizes that by taking the SAT the max number of times he can (48) before a set date (graduation), his chances of guessing a perfect score go up, due to the fact that he has taken it 48 times. We both recognize that each test attempt is an independent event and doesn't affect the outcome of another, so each individual test attempt still has the same probability of guessing a perfect score. However, he believes that by increasing the number of trials, the chance of getting a perfect score is greater.

I say that because each event is independent, each time is the same exact probability and does not change the more that you take it.

The way that makes the most sense for me to think of it is imagining 100 locks and 100 keys. You pick 1 key and 1 lock (1 test, and 1 correct set of answers). After each attempt the locks are randomized again, just like each subsequent test and correct set of answers would become random each time. Each time you try the locks (or the test), you would have a 1:100 chance regardless of how many times you try.

Which one of us (if either) is correct? And how do we go about solving this problem? Thanks in advance for the help!

Answer 1

It is true that the probability goes up with more attempts. An essentially similar problem is: rolling a 6-sided die, are you more likely to roll a 6 (the die's "perfect score") at some point if you roll it once or if you roll it 48 times?

Answer 2

Let's use a simple analogy. Suppose you have a fair die numbered from $1$ to $6$ inclusive. If you roll the die once, the chance of getting a $6$ is $1/6$. If you roll it a second time, the chance of getting a $6$ on the second roll is again $1/6$--the die does not "remember" what it rolled in the past. Each time you roll the die, the outcome is independent of any previous rolls. For any given roll, the chance of getting a $6$ remains $1/6$.

However, if the goal is to keep rolling until you get a $6$, then it clearly makes sense that the more times you roll, the more likely you are to eventually get a $6$. For example, if you rolled only once, the chance is $1/6$. But if you get to roll twice, then the chance that you get $6$ at least once is $$1 - (1 - 1/6)^2 = \frac{11}{36} > \frac{1}{6}.$$ If you get to roll three times, this chance increases to $$1 - (1 - 1/6)^3 = \frac{91}{216}.$$ If you get to roll $48$ times, this chance goes all the way up to $$1 - (1 - 1/6)^{48} = \frac{22448704993675756739157855502454784191}{2245225770735455724008721112379267481 6} \approx 0.999842.$$ So the more rolls you get to take, the higher your chance of observing at least one $6$, even though for any given roll, the chance of a $6$ remains $1/6$.

The same principle applies in your question. Even though each test is an independent trial, the more trials you are allowed to make, the greater your chance of achieving success for one or more trials, even though the individual chance of success for any given trial is the same.

That said, the chance of a perfect score by random guessing on a multiple choice exam with more than a handful of questions is extremely small. If the chance of guessing correctly on any individual question is $1/5$, then for a $40$ question test, the chance of a perfect score is $(1/5)^{40} \approx 1.099511627776 \times 10^{-28}$. That is so vanishingly small that even with $48$ trials, the chance of at least one perfect score result is still negligible, only about $5.27766 \times 10^{-27}$. In fact, for just $40$ questions on the test, you'd have to take the test approximately $10^{26}$ times just to have a nontrivial chance of guessing correctly on all of the questions.