Can someone clarify this doubt of mine, regarding calculation of average for a probability distribution?

Question

Suppose there is a question on probability which asks us to find the expected number of successful attempts or average number of successful attempts, with the probability of success given. On computation using the formula for expectation (average), the result is a mixed number, having integer and fractional part. Now, we know that the number of attempts has to be a whole number to be physically realizable. So, if the computation of mean using the formula yields a mixed number, do I round it off to the nearest integer or keep it as it is ? For instance, suppose the average number of successful attempts comes out to be 2.5. Do I mark the answer as 2.5 itself or mark it as 3 ?

Answer 1

No, you do not round the average.

For example, let's say I flip a coin and write down $0$ if I get heads and $1$ if I get tails. Now, clearly, the expected value of what I will get is $\frac12$, right?

What may be confusing you is that in every single experiment, I will get an integer value, i.e. $0$ or $1$. So how can I "expect" to get $\frac12$, when I clearly cannot?

The answer is that you don't expect to get $\frac12$ in any single throw of the coin. What you need to think about are repeats of this experiment. So, say I repeat the experiment $11$ times, then I might end up with a sequence like $$1,0,0,1,0,0,0,1,1,1,1$$

and the average of this sequence is $\frac{6}{11}$ which is not an integer, but it is pretty close to $\frac12$. And that, not the result of a single experiment, is what the average value tells me.

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Remember, the mean (also called expected value) is not

"the value I predict will come up in my experiment",

it is more like

"the average of several values I will come up in my experiment".