Verifying an answer to a probability question

Question

How do you know your answer to a probability question is correct without checking the solutions?

For example, when you have an equation you can get a graphing calculator, create the graph for the first part of the equation, the graph for the second and then get the intersection to make sure your calculations are correct.

However, how would you do that with a probability question (assuring it is correct I mean)?

For example:

At a teachers' room there are 27 lockers, arranged in the rows of 9 lockers each. (Imagine a 3 squares by 9 squares grid)

Mary, Caroline and Anthony are three of the teachers who have lockers. If those lockers are distributed randomly by the teachers, what is the probability that those 3 teachers get their lockers on the bottom row (in any position of that row)?

What I want:

• Try solving this problem and then using some simple method to assure your answer is correct.

Answer 1

I answered the earlier one, I'll answer this one too.

See, the bottom row contains nine lockers. The favourable case consists of choosing three lockers from these nine. So $\binom{9}{3}$ is the number of favourable cases.

The total number of cases is $\binom{27}{3}$. So the answer would be $\dfrac{\binom{9}{3}}{\binom{27}{3}}$.

I cannot genuinely find any method of verifying your answer, except a few heuristics.

1) Work over the problem carefully, so that you don't need a re-check. If needed, work it out in rough, and then polish the argument.

2) When you are verifying a formula, always see to it that it applies in small cases. If you want, try it out for $6$ lockers and $3$ rows, or $3$ lockers and $3$ rows, and see if you can spot a pattern in your working.

3) If you are utterly not convinced with your answer, ask a friend to write a program, and simulate it, or simulate the algorithm in your head if you can. This may help in larger situations and when you have more time.

4) Your other help is to take a course that facilitates basic rigour, like real analysis. Once you do this, your confidence while taking on probability will sky rocket, because this promotes the step-by-step careful consequential thinking that probability demands.

Aside from this, I am not sure I can say more, because for me, this has worked (I read Rudin for Real analysis). The problem with probability is that you can't go backwards and verify your answer, neither can you have many easy ways of approaching the answer. Hence, this answer is simply my take, but if I'm successful, I think you can be too.

On that note, best of luck with probability. It's a good journey.

Answer 2

I doubt I will be allowed to do that during an exam

If you are having doubts with answering a question of this simplicity in an exam, you may not pass the exam due to nervousness.   Have a little more confidence in yourself.

This sort of problem has just two stages.

• (1) Interpret the question.   This one is only asking for the probability for selecting three from nine of the twenty-seven lockers.

• (2) Calculate it.   That will be just the ratio of two binomial coefficients using those numbers.

There is not much you can actually do wrong with this kind of problem; it is just testing reading comprehension and basic knowledge.   Recognising what you need to do is the trick; then you either know how to do it, or you don't.

You can't really verify the results, so just answer the question as best you can, and move on.   Don't let doubts fog your thoughts.

Answer 3

Assuring (making sure) a solution is correct is a philosophical problem really, but I know you are not interested in this. You basically want to validate your solution, or in other words increase your confidence that your solution is correct.

A generic way to validate probability problems is to run simulations that model the problem at hand and simply count the number of favourable events and divide it by the number of total efforts to get an estimate of the probability you are interested in. This usually involves writing small programs. In an exam setting this is out of the question, but you can do it when you are studying to validate some solutions.

Another generic way to validate results is to think of different ways of solving the problem and see whether your solutions agree. Trying to think of the problem from different angles can get you a long way.

In this particular problem you can think in terms of combinations as the other answers suggested, and you can also think about it in terms of probabilities of successive choices. So the probability of the first teacher being assigned a locker at the bottom row is $9/27$. Then for the second teacher we have $1$ locker less so the probability of getting a bottom locker is $8/26$. Finally the third one has $2$ lockers less available so the probability is $7/25$. $$\frac{9}{27}\cdot \frac{8}{26}\cdot \frac{7}{25} = \frac{9 \choose 3}{27 \choose 3} \approx 2.87\%$$

Finally, let me point out that doubts often come because we have not grasped the material very well. Maybe we have not built the right insights. So study hard to gain understanding. The more at ease you are with the notions at hand the more confident you will feel about your first solution in an exam. But while studying, do try to think about the problems in different ways. It will build your intuition and understanding. Let me give you a minor example that just happened to me while answering this question. When I tried to answer the question based on the probabilities of consecutive selections my first answer was $\frac{9}{27}\cdot \frac{8}{27}\cdot \frac{7}{27}$. Stupid mistake you might say, but I did not see it immediately. I noticed the solutions did not agree and I was puzzled. It did not find the mistake immediately. I had to simplify the combinations fraction to see the pattern and then go "of course!". Now I am less likely to do the same mistake again :)

Have a great time in your learning journey!