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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.

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  • $\begingroup$ 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. $\endgroup$
    – Math1000
    Oct 24 '19 at 2:28
  • $\begingroup$ Yes, expected value is linear: see here. $\endgroup$ Oct 24 '19 at 2:29
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Hint: For each roll, the expected outcome is $$(\textrm{net result for }1) \cdot P(1) + (\textrm{net result for }2) \cdot P(2)$$

You should be able to compute this explicitly.

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.

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  • $\begingroup$ So i'll win $5000\$$ ? $\endgroup$ Oct 24 '19 at 2:33
  • $\begingroup$ “Will win”? No. But if a large number of people roll 100 times each, the average winnings among all these people will be approximately 5000. No specific outcome is ever guaranteed. The expected value is like a long-term average of repeated experiments. If you roll 100 times, it is possible (but not likely) that you will lose every single time. Just like if you flip a coin 100 times, you might get heads every time, but this is not likely. $\endgroup$
    – MPW
    Oct 24 '19 at 2:46
  • $\begingroup$ Thanks for that explanation. $\endgroup$ Oct 24 '19 at 2:55

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