# Is the expected value independent of the number of repetitions?

I have this statement:

Some game of throw a dice with two equiprobable sides, with the number $$1$$ and $$2$$. If you get $$1$$ You win $$200\$$ and if you get $$2$$ you lose $$100\$$, after $$100$$ launches what value is expected to win?

My attempt was:

I understand that expected value is a theorical concept that give us an averague value of the experiment, something like the average, but with probabilities.

For this exercise i should calculate the expected value, but my doubt are:

i) Is my theorical concept correct?

And more important, to solve this exercise:

ii) The expected value applies by independent to each repetition of the experiment?, i.e when i go to calculate the expected value i should multiplicate the expected value by the number of repetitions?. Per example if i play this game $$1$$ time i hope to win $$50\$$ but if i play this game the $$100$$ times i should hope The same money as if I played it once?, i.e $$50\$$ Or multiplied by the times I will play? in this case 100 times $$50\ \cdot 100 = 5000\$$ ? Thanks in advance.

• Assuming each game is independent, then each time you play, you expect to win $1/2\cdot 200 - 1/2\cdot 100 = 50$ dollars. It doesn't matter how many times you play. Oct 24 '19 at 2:28
• Yes, expected value is linear: see here. Oct 24 '19 at 2:29

Hint: For each roll, the expected outcome is $$(\textrm{net result for }1) \cdot P(1) + (\textrm{net result for }2) \cdot P(2)$$
Since the results for each roll don’t depend on previous results, the expected outcome for $$n$$ rolls is $$n$$ times the expected outcome for a single roll.
• So i'll win $5000\$\$ ? Oct 24 '19 at 2:33