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Probability and expected value

You are in business and you are given two opportunities. The first you will have to invest $5000 but you could earn $25,000 less your initial investment. The second you only have to invest $2000 but then you will only earn $10,000 less your investment. The probability of getting $25,000 opportunity is 3/10 while the probability of getting $10,000 is 2/5. Which should you choose based on expected values?

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George P. | Mathematics: High School/Undergrad/Actuarial exam preparation.Mathematics: High School/Undergrad/Actua...
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Hi Teresa. For two-case probability and expected value type questions like this, we are calculating how much we can expect to make in the future, based on two variables - magnitude of the opportunity and the likelihood that it will occur.  For this particular question, we look at the formula for expected value, E[x], given the probability, p(x), of earning x dollars:

E[x] = p(x)*x

For example, if I flipped a fair coin and promised you $20 if a heads came up, you can expect to earn:

E[x]= (.5) * ($20) = $10    (Notice .5 is the probability of a heads facing up)

Now for your example, there is a slight catch; for each opportunity, you are required to invest an initial amount (there is a 100% chance that you pay this initial amount) or E[x]=(1)* (initial_investment) = - initial_investment (notice the negative sign because you are paying, not earning, this amount).


After calculations, we see E2[x] > E1[x], and we would rather choose Opportunity 2 because it is expected to earn a larger sum of money.


Unless I'm interpreting the question wrong, shouldn't opportunity 1's EV be:

(3/10)*25,000 - 5000 = $2500

and opportunity 2's EV: 

(2/5)*10,000 - 2000 = $4000

Because when opportunity 1 pays off, we get $25000 (but $5000 of that was our initial investment) and when opportunity 2 pays off, we get $10K (but $2k of that was our own)

- that's the way I interpreted the "you could earn $25,000 less your initial investment" and "you will only earn $10,000 less your investment" but maybe I should be reading it as "you get $20,000 for opp. 1 [and $8000 for opp. 2] at the end of the investment", as in the instituition (or whomever) is keeping our initial investment.