non positive probability

Question

The probability of getting 1 on a biased die is a non-positive number. For all the other numbers upto 6, the probability is same. What is the probability of getting an odd number on throwing this die? I found this question in an exam paper. Can probability be really non positive? and How should we approach this question?

Answer 1

The probability axioms directly imply that in any probability space, every event $E$ will satisfy $0\leq \Pr(E)\leq 1$. In particular, probability in a probability space is never negative.

That said, the concept of negative probability does exist, but this would not be in a probability space but rather would be in a quasiprobability space. It is safe to completely ignore the existence of quasiprobability spaces. You will likely never need them, and you need only consider them when you very explicitly know you need them for something very specific.

For your particular problem, it is clear from context that we are talking about traditional probability, as is going to be the case throughout your entire academic career. As such, being told "the probability of rolling a $1$ is non-positive" and remembering that as per probability axioms every probability is greater than or equal to zero... this implies (as pointed out in the comments) that the probability of having rolled a $1$ must be identically zero. Make sure you understand that the negative numbers are in reference to $\{x\in\Bbb R~:~x<0\}$ and non-positive numbers are in reference to $\{x\in\Bbb R~:~x\leq 0\}$. The sets are similar however the non-positive numbers also include zero, unlike the negative numbers. Recall that zero is neither positive nor negative.

From this point, we learn that since the rest of the outcomes on the (presumably six-sided) die are equally likely and that they account for all remaining possibilities, in order to have their respective probabilities be equal and all add up to $1$ it must be the case that the probability of rolling an $i$ is equal to $\dfrac{1}{5}$ for each $i\in\{2,3,\dots,6\}$.

Now, from here, we ask what the probability is of having rolled an odd number on this die. Well, that corresponds to having rolled a $1$, a $3$, or a $5$. Since rolling a $1$ occurs with probability zero, and rolling a $3$ occurs with probability $\dfrac{1}{5}$ and rolling a $5$ occurs with probability $\dfrac{1}{5}$, adding these together gives us our final answer of

$$\dfrac{2}{5}$$