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I am a little bit confused concerning the following problem.

Lets suppose you have 10 multiple choice questions which can be ticked as true or false. For each correctly crossed answer you receive $+1$ point otherwise you get $-1$ point.

It is not possible that you end up with a negative amount of points e.g. if you had accumulated $-4$ points at the end of this multiple choice exercise the amount is set to $0$.

What is the expected value (ev) if you randomly tick true or false?

My approach:

Without having done further calculations I would say that the ev must be $>0$. The sample space consists of 10-tuples and each tuple has the same probbility (Laplace experiment). Calculating the ev means that you multiply the probability of a tuple with the amount of points this tuple is associated with and then you take the sum over all this multiplications. As the result of each multiplication is $\geq 0$ and you have at least one tuple with an amount of points $>0$ the ev must be greater than $0$.

Philipp
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  • As you say, the expectation is obviously positive: no test result can be negative and some of them are positive. – lulu Nov 05 '19 at 12:34
  • Just do the sum mechanically. Letting $X$ be the number of correct answers use the binomial distribution to write out the probability distribution for $X$. Use that to get the probability distribution for the score. – lulu Nov 05 '19 at 12:35
  • What is wrong if I consider each question on its own and conclude that the expected value is 0 for each question hence the expected value for answering all questions randomly is 0 as well? – Philipp Nov 05 '19 at 12:47
  • Because the expected value on the test is not the sum of the expected values of each score. It is the minimum of that sum and $0$. – lulu Nov 05 '19 at 12:49
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    To clarify your thinking, solve the same problem for a two question test. In that case your four possible raw scores are ${-2,0,0,2}$ where each outcome has probability $\frac 14$. That means that the final score can be ${0,0,0,2}$ again with probability $\frac 14$ for each state. That gives a final expectation of $\frac 24=\frac 12$. Not $0$. – lulu Nov 05 '19 at 12:52

2 Answers2

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Guide:

Let $X$ denote the number of questions that are answered correct.

Then the number of points at the end can be written as $f(X)$ where $f$ is the function that maps the number of questions correctly answered to the associated amount of points.

The last step is determining:$$\mathbb Ef(X)=\sum_{k=1}^{10}f(k)P(X=k)$$

drhab
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You have $2^{10}$ possible outcomes.

Let's see when you get a non-zero score:

Score 2k) You need to answer 5+k right, 5-k wrong. The different possibilities are${10}\choose{5-k}$.

Thus expected value is thus

$$\frac{1}{2^{10}}\sum_{k=1}^5 2k {{10}\choose{5-k}} = \frac{1260}{1024}\simeq 1.21 $$

Alberto Saracco
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