Probabilities associated with negatively marked questions

Question

First of all: not a native english speaker, and not a mathematician.

Please explain as you would to your 10 years old son.

• I have 120 questions to answer True or False
• For each right answer, i score 1 point
• For each wrong answer, i lose 1 point
• Not answering a question does not affect the score (=0)
• I don't know if the T/F is evenly distributed, so it could be 50/50, 85/15, etc (%)
• I don't know the answer for any of them, so i'll just guess them all

Given the above, can i say that:

1. I have 50% chance of getting each question right?
2. I have 50% chance of getting all questions right?
3. Answering all is better than answering just some of the questions?
4. I have to score a minimum of 70 points, but only know the correct answer for 50 questions; how many others would be "safe" to guess? Only 20? The remaining 70? Or something in between?

Feel free to edit the question (or the title) if i wrote something wrong.

Answer 1

1. If you guess randomly, half chance T and half chance F, then yes.

2. No. You have 50% chance of getting the first question correct, and then only 50% chance of getting the second question correct, and 50% chance each for the remaining 118 questions. Even you get the first correct (50% chance), your chance of getting the first two correct is only 50% out of the first 50%, i.e. 25%. Then, for your third question, you have only 50% out of the 25% chance to have all first three questions correct, i.e. 12.5%.

   Therefore, when you keep calculating, you have $0.5^{120}$ chance of getting all questions right, which is very unlikely.

3. If you answer/guess a question instead of leaving it out, you have half chance of losing a mark but also half chance of getting a mark. Therefore, you expect not to get any marks on average if you repeat this for many times. Whether to take this risk is entirely personal up to your risk preference; most people prefer making less risky choices if the expected outcomes are the same, but again this is up to your preference.

4. If you only know 50 questions, then you still have no idea on the remaining 70 questions. Similar to Q3, if you guess for 20 more questions, you have a chance of getting a pass, but you still have a greater chance to get 19 or less. (Why? the reasoning is similar to in Q2) And if you guess all 70 questions, your chance of getting 20 more marks in fact does get smaller (after deducting those questions you guessed wrongly), because you will have to guess at least 45 questions correctly to score 20 more. Statistically, you have no chance (far smaller than 0.01%) of getting a pass, no matter you choose to guess 0 or 20 or 70 questions.

Answer 2

1.) Yes, you have a $50\%$ chance of getting each question right, as for each question, there is one right answer out of two answers total, and so if you pick at random, there is a one in two, or $\frac{1}{2} = \frac{50}{100}$ chance that you pick the correct answer.

2.) No, you do not have a $50\%$ chance of getting all questions right. For simplicity, consider the case where you only have two questions (Call them A and B.) Now, there is a $50\%$ chance that you will get question A right. However, even if you get question A right, there is still only a $50\%$ chance that you will get question B right. Therefore, there is a $50\% * 50 \% = \frac{50}{100} * \frac{50}{100} = \frac{2500}{10000} = 25\%$ chance of getting both questions right. As the number of questions goes up, this probability will decrease even further, as you have to 'get lucky' in picking the right answer more and more times.

3.) Answering any number of the questions is expected to give you the same result. Your choices are between answering and not answering for any given question.
First, consider the case of not answering. In this case, there is a $100\%$ chance of getting $0$ points, and so if you do not answer a question, you will expect to get, on average $\frac{100}{100} * 0 = 0$ points.
Now, consider the case of answering. In this case, there is a $50\%$ chance of getting $1$ point, and a $50\%$ chance of getting $-1$ point. Therefore, on average, you will expect to get $\frac{50}{100} * 1 + \frac{50}{100} * -1 = 0$ points from answering a question. While it is impossible to get $0$ points from answering a question, this is more saying that in the long term, it is likely that you will answer as many questions incorrectly as correctly, and so their point values will cancel out, and on average, your score per question will be $0$ regardless of how many questions you answer or do not answer.
As your average total score is the number of questions times the average score per question, your average total score will also be $0$.

4.) If you know the answers to $50$ questions, you can still use the analysis from part $3$ to see that there is no advantage or disadvantage to guessing on the remaining $70$ questions. If you guess, on average, you will break even on them. However, it is also true that outside the realm of probability, it may be more advantageous to do one thing or the other. Maybe you know that a $50$ is too low of a score, and so it's worth the risk of getting less than a $50$ in order to have the chance to do better. In this case you should guess. Alternately, maybe you know that a $50$ is a good score, and so you'd rather not risk having it go down. In this case, you should probably not guess. In something like this, where there's no advantage on average, but one option increases the range of scores you can have, the main thing that determines how many questions you guess on is how risk-averse you are. The more risk-averse, the fewer questions you will answer, so as to not lose what you already have.

If there's anything unclear in here, please ask, and I can try to clarify.